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Induction theorems for finite groups including a common generalization of two classical theorems
Journal of Algebra 528 (2019) 217–230
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Journal of Algebra www.elsevier.com/locate/jalgebra
Induction theorems for finite groups including a common generalization of two classical theorems William F. Reynolds 1 Department of Mathematics, Tufts University, Medford, MA 02155, USA
a r t i c l e
i n f o
Article history: Received 28 February 2018 Available online 16 March 2019 Communicated by Leonard L. Scott, Jr. MSC: 20C15 Keywords: Induction theorem Character ring K-elementary group p -section π -section Ring of functions Idempotent
a b s t r a c t Roquette proved results involving induction about rings related to the character rings of finite groups and consisting of functions that vanish outside a p -section; Kletzing did something similar for Q-characters. This paper gives new generalizations of these results involving K-characters for any field K of characteristic 0 as well as corresponding theorems for a set of primes. That these facts are useful is illustrated by showing that they imply a common generalization of the Witt-Berman and Artin induction theorems as well as results of Chen, Fan, Hu, Robinson, and Sin. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The best-known proof of Brauer’s induction theorem is probably that of Brauer and Tate [3]. Extensions of this proof to the Witt-Berman generalization over a field K of characteristic zero are found in [8, (21.6)] and [22, §12.6, Theorem 27 and Proposition 36]. These proofs use results about certain functions on a group G that vanish outside
1
E-mail address: fi[email protected]. Current address: 3 Preble Gardens Road, Belmont, MA 02478, USA.
https://doi.org/10.1016/j.jalgebra.2019.03.009 0021-8693/© 2019 Elsevier Inc. All rights reserved.
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a K-p -section of G for a prime p; this set of functions depends on a coefficient domain D. (Some terms are defined later.) In the case that K = C and D is p-adically complete for some p dividing p, Roquette extracted such results from earlier proofs and expanded them; in this way he reduced to a situation involving K-p-elementary subgroups (p-elementary in his case). Kletzing proved a similar result with K = Q, not assuming completeness. The aim of this paper is to present a generalization of Roquette’s and Kletzing’s reductions for arbitrary K, then a corresponding result for sets π of primes, and to demonstrate the power of these results by presenting a common generalization of the Witt-Berman and Artin induction theorems, thus strengthening a result of Chen and Fan that deals with the case that K = C. More specifically, let x be a p-regular element of G, let D = Z[ωo(x) ]∩K and Dp = Zp D where Zp consists of the p-integers in Q, and let R be the algebra of K-characters of G. A preliminary result, Theorem 3.1, states in effect that the characteristic function ε of the K-p -section S of x, an idempotent, is in Dp R. Thus Ap = (Dp R)ε is a direct summand—in fact indecomposable—of Dp R. Furthermore ε is induced from an element where R is the analogue of R for a K-p-elementary subgroup G = xP with P of Dp R as large as possible. Theorem 4.1 improves the induction statement and leads to Theorem 4.2, a reduction to K-p-elementary subgroups which is something like a Witt-Berman theorem on K-p -sections. This says, among other things, that Ap is induced bijectively p of Ap for G and that there is a restrictionfrom a certain subalgebra of the analogue A like isomorphism in the opposite direction. For K = C a connection between Roquette’s results and Thompson’s characterization of characters is pointed out. In Section 6, p is replaced by a set of primes and an induction theorem involving K-π -sections is proved. The common generalization is in the final section. My theorems also generalize or imply some results of Fan, Hu, Robinson, and Sin; this is evidence that they have other uses although they are rather technical. The proofs, like classical proofs of induction theorems, follow a pattern of first constructing a suitable function by induction and then replacing it by a constant times an idempotent. I have chosen to base the arguments for replacement on modifications of a lemma of Banaschewski, Goldschmidt, and Isaacs about functions on finite sets; these modifications are given in Section 2. By applying these instead of a well-known argument involving powers of functions, the intuitiveness of the proofs is arguably improved at the cost of using more standard facts from algebraic number theory. I never assume any form of Brauer’s theorem. 2. Functions on a finite set The lemmas in this section were developed from a lemma [12, Lemma 3.2], [14, Lemma 8.5], [8, Lemma 15.11] that can be stated in contrapositive form as follows: Lemma 2.1. (Banaschewski-Goldschmidt-Isaacs) Suppose that C is a “rng” (ring without identity assumed, or Z-algebra) of functions from a nonempty finite set T to Z such that
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for every t ∈ T and every prime number p there exists ψ ∈ C such that p ψ(t). Then the 1-function 1T is in C. This follows easily from a result of Banaschewski [1, Proposition 1] on maximal ideals of rings of functions. (Hint: choose the ring in [1] to be generated by 1T and the “rng” in [12]). Goldschmidt and Isaacs attributed this result to Banaschewski but I think that it is different enough from his result to bear all three names, especially because it has a simple direct proof. Lemma 2.1 can be strengthened in three ways and then weakened (for simplicity) in one way to yield the next result. Let T be a nonempty finite set and R a commutative ring (with identity 1R ). For any ψ in the R-algebra F(T, R) of functions from T to R and any ideal p of R, let ε be the characteristic function of Sψ,p = {t ∈ T | ψ(t) ∈ / p}
(2.1)
in F(T, R). Sψ,p may be thought of as the support of ψ mod p. (Gluck [11, p. 126] has also used a modification of Lemma 2.1.) Lemma 2.2. For T and R as above, let ψ ∈ C where C is an R-subalgebra (no identity required) of F(T, R), i.e., an R-submodule that is closed under multiplication. Let p be a prime ideal of R. (The zero-ideal is allowed if it is prime, namely if R is an integral domain.) Then for ε as above, rε ∈ C for some r ∈ R, r ∈ / p. If Sψ,p is the same set for all p in a set Π of prime ideals, then the same r can be taken for all these ideals, so that r∈ / p for all p ∈ Π. Proof. Let S = Sψ,p and S = T \ S. First a “clearing” step, showing that we can replace ψ by some θ ∈ C with Sθ,p = Sψ,p such that θ vanishes on S . Taking S = ∅, define θ = ψ· s ∈S ψ−ψ(s )1T ; if we multiply θ out and look at the terms, we see that θ ∈ C; the factored form of θ(t), t ∈ T , shows that Sθ,p = Sφ,p since p is prime; the vanishing is obvious. Now expanding the equation s∈S θ(s)ε − θ = 0 proves the lemma since r = θ(s) ∈ / p; this is a “leveling” step replacing θ by rε. (If S = ∅, then θ = 0 = ε so that r = 1 works.) The last sentence of the lemma is true since the construction of r is independent of p for p ∈ Π. For use with some results such as Lemma 5.2 and Theorem 7.2, whenever I use the word “prime” as a noun I shall allow the possibility that the prime is 0; cf. [16, pp. 115, 140]. (Alternatively, to avoid unconventional terminology, I could use prime ideals of Z, including 0Z, instead of primes.) Definitions can be written so that they do not treat 0 individually, although the proofs may; I omit details. I shall not always give proofs for p = 0. Some algebraic number theory can give a constant in Z:
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Lemma 2.3. Given T , assume that R is the set D of all the algebraic integers in an algebraic number field F . Let ψ and C be as in Lemma 2.2 with ψ ∈ C and let p be a prime. Suppose that Sψ,p is the same set S for all p in the set Π of the prime ideals of D that contain p, except that if p = 0 suppose that S = Sψ,0D . Then kε ∈ C for some positive integer k, p k, where ε is the characteristic function of S on T . / p whenever p ∈ p. Proof. If p = 0, then Lemma 2.2 gives d ∈ D such that dε ∈ C while d ∈ We can replace d by a positive integer as follows. D is a Dedekind domain [8, pp. 76–78], [15, Theorem 3.16]. Hence if n is the norm of d from F to Q, then straightforwardly n ∈ Z, d = 0, n/d ∈ D, and p n (see [15, pp. 18, 20, 24, 42–44]); take k = ±n. Change this slightly if p = 0. (It is probably well known that [3, Theorem 3] can be proved by applying Lemma 2.1 to η there. Isaacs [14] used the same lemma differently in proving Brauer’s theorem.) 3. Induction from a K-p-elementary subgroup G will always be a finite group and K a field of characteristic 0. For any n let ωn be a primitive n-th root of unity in some extension of K. Let h be any multiple of the exponent e(G) of G and let L = K(ωh ). Then there is an action of the Galois group Gal(L/K) on G where each element acts on every cyclic subgroup g of G in the same way that it acts on ωo(g) , where o(g) is the order of g. The orbit g K of g (hence everything we care about) is independent of the choice of h; thus we can use the same L for subgroups of G. Combining this action with conjugation gives an action of G × Gal(L/K) on G, whose orbits are the K-conjugacy classes or K-classes g G,K . For any p -element (p-regular element) x of G, let xG,K,p be the K-p -section of x in G, i.e., the set of all elements g of G whose p -part gp is in xG,K ; similarly for subgroups. For any prime p I shall say that a group H is K-p-elementary or K-elementary for p if H = xP where x is a p -element and P a p-group that normalizes xK [8, p. 492], [22, §12.6]. Here the prime 0 behaves like any prime that does not divide |H| (or e(H)); in particular, “K-0-elementary” is the same as “cyclic.” Results for p = 0 are often trivial, but not always. Let R(KG) be the ring of K-characters of G and similarly for other groups and fields; I abbreviate R(KG) by R. For any integral subdomain D of K, the irreducible K-characters of G form a D-basis of the D-algebra DR (see [8, pp. 207, 493]), [22, §12.1, Proposition 32]) and a K-basis of the K-algebra of K-valued K-class-functions [8, (21.3)], [22, §12.4, Corollary 2]; hence the latter is KR. Thus DR can be identified with D ⊗Z R, etc. For any algebra J of functions with values in an extension of D, let the set of all functions in J with values in D be called JD . (This notion will be used to apply Lemmas 2.2 and 2.3.) For any union U of K-conjugacy classes of G, let εU = εU,G be the characteristic function of U on G, an idempotent, and similarly for other groups; thus εU ∈ KR.
(3.1)
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The following notation will be used throughout. Notation. Given any G, K, p, and a K-p -section S of G, choose a p -element x of G such that S = xG,K,p . Let N be the normalizer NG (xK ) in G of the K-class xK . Choose = xP , so that G is K-p-elementary. (The any p-Sylow subgroup P of N and let G choices of x and P do not really matter.) Let D = Z[ωo(x) ] ∩ K.
(3.2)
D consists of all the algebraic integers in Q(ωo(x) ) ∩ K [15, Theorem 10.4], [8, (4.5)]. Of course D depends on the section. Define ε = εS,G , ε = εS, G . Denote restriction and induction between G and G by Res and Ind without the usual indices and let S = xG,K,p , (⊆ R). = R(K G), V = Ind R R By the equation (Ind α)β = Ind(α Res β)
(3.3)
[8, (10.20)], [22, §7.2, Remark (3)], V is an ideal in the ring R and hence a Z-algebra. Most of my results carry over for suitable larger choices of D; the arguments throughout the paper would be somewhat simpler for Z[ωe(G) ] or Z[ω|G| ] as in references above but lead to weaker theorems. I have chosen domains that yield what seem to be natural forms of the results. The D of (3.2) can also be useful in special cases, e.g., if K ⊆ R. The proof of Theorem 3.1 follows the pattern described in the introduction: first I shall carry out an induction argument by modifying lemmas from some proofs of the Witt-Berman theorem, taking [8] and [22] as primary references (similar arguments can be found in [10] and [23], say); then I shall apply Lemma 2.3 to make a replacement argument. Lemma 3.1. Let H = xP be K-p-elementary. Then every K-character (i.e., trace of a representation over K) of x is the restriction of a K-character of H with values in Z[ωo(x) ] ∩ K. Proof. The restriction statement is in [8, (21.8)] and [22, §12.7, Lemma 17]. The value statement can be read off from [23, Lemma 2]. D such that Lemma 3.2. For any K-p -section S of G, there exists ξ ∈ (DR)
ResG x ξ = o(x) εxK ,x .
(3.4)
each with Proof. This is the same as [8, (21.9)] for G and [22, §12.7, Lemma 18] for G, minor adjustments and use of Lemma 3.1. (If K = C, as for Brauer’s theorem, then = x × P and we can replace D by Z in this lemma and below (see [3, §5, I]).) G
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D is such that ResG ξ = Lemma 3.3. For any K-p -section S of G, suppose that ξ ∈ (DR) x aεxK ,x for some positive integer a with p a. Let ψ = Ind ξ. Then (a) (b) (c) (d)
ψ ∈ (DV)D ; ψ(g) = 0 if g is a p -element of G not in xG,K ; ≡ 0 (mod p); ψ(x) = a|N : G| if also a = o(x) as in (3.4), then ψ(x) is the p -part of |N |.
Proof. This follows from the proof of [8, (21.11)] or [22, §12.7, Lemma 18]. Lemma 3.4. For any G, K, and p, let E be an integral domain included in Z[ωe(G) ] and suppose that p is a prime ideal of E such that p ∈ p. If θ ∈ (ER)E (the superscript being like D above) and if g and g are in the same K-p -section of G, then θ(g) ≡ θ(g ) (mod p). Proof. Apply [8, (21.12)] to get the corresponding result for Z[ω|G| ] or use [22, §12.7, Lemma 16] similarly for Z[ωe(G) ]; then reduce to E using [15, pp. 28–29]. Lemma 3.5. Under the assumptions of Lemma 3.3, ψ(g) ≡ a|N : G|ε(g)
(mod p)
(3.5)
for all g ∈ G and for all prime ideals p of D such that p ∈ p (recall that ε = εS,G ). Then Sψ,p = S for all such p if p = 0 and for p = 0D if p = 0. If also ψ has values in Z, then Sψ,pZ = S. Proof. (3.5) is immediate from the preceding two lemmas; the function ψ corresponds to ψj in [8, p. 498, Step 1] and to ψλ in [22, §12.7, Lemma 19]. By (2.1) the other statements follow since ψ(x) ∈ / pZ = p ∩ Z. Finally Lemma 2.3 comes into play to give a result involving the K-p-elementary group G. Theorem 3.1. For any K-p -section S of G, ⊆ DR kε ∈ DV = Ind DR = DR(K G) with D as in (3.2). for some positive integer k, p k, where DR Proof. In the hypothesis of Lemma 2.3, take T = G, D and p as given, ψ = Ind ξ with ξ given by Lemma 3.2, and C = (DV)D , a D-algebra. By Lemma 3.3(a), ψ ∈ C. Then Sψ,p = S for p as in Lemma 3.5; hence Lemma 2.3 applies and gives the result. (For p = 0 this theorem can be obtained more directly from Lemma 5.2 below.)
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As in the introduction, Theorem 3.1 can be stated in terms of a semilocal integral domain; see the next section. It will be improved by Theorems 4.1 and 5.2. 4. Induction from a K-p -section In this section we make the set of inducing functions in Theorem 3.1 smaller and then localize the coefficient domain to obtain Theorem 4.2. Lemma 4.1. For any K-p -section S of G, every element of S is conjugate in G to an ˜ element of S. Proof. (Cf. [21, (26)].) Replacing s ∈ S by a conjugate in G, we can suppose that sp ∈ xK . Then the p-part of s is in some p-Sylow subgroup Q1 of C := CG (x) and s ∈ xK Q1 . We can replace the new s by a conjugate in N (= NG (xK )) to get a third and sp ∈ xK s ∈ xK Q with Q included in the p-Sylow subgroup P of N , whence s ∈ G (It follows that Q = P ∩ C.) so that s ∈ S. For any algebra J of K-class-functions on a group G and any subset T of G, let [J|T ] consist of all those functions in J that vanish outside T . Let S be a K-p -section of G. By (3.1), ε ∈ KR; hence F := (KR)ε = [KR|S] is a := (K R) ε= ring with identity ε and also a direct ideal summand of KR; similarly for F [K R|S]. For φ ∈ F define ResSS φ = (Res φ) ε ∈ F.
(4.1)
This modification of restriction is similar to [21, (29)] and [16, pp. 145-146]. By a statement preceding (3.1), F is the set of all K-class-functions from G to K that vanish outside S, with {εsG,K | s ∈ S} as a K-basis. ResSS εsG,K = εsG,K ∩S , which is nonzero by Lemma 4.1. Thus a K-basis of ResSS F is given by the distinct characteristic such that θ has G denote the set of functions θ ∈ F Let F functions εtG,K ∩S , t ∈ S. the same value on any two elements of S that are K-conjugate in G, with corresponding notations for similar situations. (Gluck [11, p. 125] calls functions like these G-invariant.) N , Ind gives a K-space-isomorphism G = F Lemma 4.2. For any K-p -section S of G, F S N to F, and Res is a K-algebra-isomorphism of F to F N . of F S G . F N since a conjugating element fixes S and hence G = F Proof. Clearly ResSS F = F
fixes xG,K = xK . The isomorphisms follow from the above description of K-bases, since the formula for induction implies that Ind εtG,K ∩S˜ is a positive multiple of εtG,K . Now I shall build on Theorem 3.1 by repeating the induce-replace pattern; this time the output of that theorem provides the input of the induction argument. Thus the entire proof of Theorem 4.1 requires going through the pattern twice.
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Define D-algebras A := DR ∩ F = [DR|S],
:= DR ∩F = [DR| S], A
N := A ∩F N . A
(There is an analogy of A to the algebra of virtual or generalized principal indecomposable characters and of B below to the Brauer character algebra: see, for example, [18, p. 239].) By Theorem 3.1 there exist positive integers k and l, not divisible by p, such that in fact, l N . Then kε ∈ DR whence kε ∈ A and similarly l ε ∈ A; ε∈A k ε = ResSS kε ∈ ResSS A =: Y,
k Ind ε ∈ Ind Y =: X.
(4.2)
Y is a D-algebra since ResSS preserves products. Consider the D-algebra B := (DR)ε; clearly A ⊆ B. If β ∈ B, then β = θε, θ ∈ DR. A ⊆ A; hence we can write lY ⊆ Then ResSS lβ = (Res θ)(Res lε) ε = (Res θ)(l ε) ⊆ (DR) S N . By induction lX ⊆ Ind A N ⊆ Ind A := W. Then ResS lB ⊆ A lX ⊆ W ⊆ A ⊆ B.
(4.3)
If γ ∈ X and δ ∈ A, then γ = Ind ζ, ζ ∈ Y, and ζ Res δ = ζ ε Res δ = ζ ResSS δ. By (3.3), γδ = Ind(ζ ResSS δ) ∈ Ind YY ⊆ Ind Y = X so that XA ⊆ X. By (4.3) lX ⊆ A, so that lXX ⊆ XA ⊆ X and lX is a D-algebra. Now to use Lemma 2.2: take T = G, R = D = Z, p = pZ, ψ = kl Ind ε, and C = (lX)Z . and x ∩ S = x ∩ xG.K Since l ε∈A = x ∩ xK = xK , ξ = kl ε satisfies the hypothesis of Lemma 3.3. Since ψ = Ind ξ, Lemma 3.5 gives Sψ,pZ = S. By (4.2) ψ ∈ lX, hence ψ ∈ C. Lemma 2.2 now implies that vε ∈ lX ⊆ X with p v; together with (4.3) this implies: Theorem 4.1. For any K-p -section S of G, S] ⊆ [DR|S] ∩ DV ⊆ DR vε ∈ X = Ind ResSS A and vε ∈ W = Ind[DR|
(4.4)
for some positive integer v not divisible by p. The second statement strengthens Theorem 3.1; in turn it will be made explicit by Theorem 5.2. If α ∈ A, then vα = vεα ∈ XA ⊆ X; thus vA ⊆ X.
(4.5)
N ⊆ vW ⊆ X = Ind Y and Lemma 4.2 implies that v A N ⊆ Y. Hence Ind v A To formulate the next theorem I introduce the semilocal integral domain Dp := Zp D as coefficient ring. (I have postponed doing this in order to get information about constants in Z in the next section.) For any of our D-modules M, let Mp = Zp M, a Dp -module. N )p = A p ∩ F N , Bp = (Dp R)ε, etc. By (4.3) and (4.5) Easily, Ap = Zp A = [Dp R|S], (A
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p = Ap . Since lY ⊆ A N and v A N ⊆ Y, we have Xp = Wp = Ap , i.e., Ind Yp = Ind A Yp = (AN )p . We now have the following reduction to a situation involving a K-p-elementary sub N )p = A p ∩ F N and similarly with G in place of N , this theorem can group. Since (A be stated entirely in terms of Dp -modules. A corresponding reduction in which Dp is replaced by a larger local ring Dp can be deduced easily from it. Theorem 4.2. For any K-p -section S of G, (a) ε ∈ Ap and Ap = [Dp R|S] = (Dp R)ε is an indecomposable direct summand of Dp R = p; Dp R(KG); similarly for A N )p to Ap ; (b) Ind(AN )p = Ind Ap = Ap and Ind gives a Dp -module-isomorphism of (A S S N )p ; (c) ResS Ap = (AN )p = (AG )p and ResS gives a Dp -algebra-isomorphism of Ap to (A N )p , and ResS φ ∈ A p are equivalent. (d) if φ ∈ F, then the relations φ ∈ Ap , ResSS φ ∈ (A S Proof. This is mostly a summation of the results above. In particular by Theorem 4.1, ε ∈ Xp = Ap whence Ap = Bp . The idempotent ε of Dp R is indecomposable: a proper summand of ε would have values 0 and 1; lifting it to DR and applying Lemma 3.4 gives a contradiction. Hence Ap is an indecomposable direct summand. For K = C Roquette [21, §3] proved results equivalent to almost all of this theorem with the p-adic completion of Z[ωe(G) ] instead of Dp . He took advantage of the fact that = x × P to use functions on P instead of on S = xP ; if K = C we could do he had G p = [Dp R(C G)|xP the same, shifting by the isomorphism of A ] to Dp R(CP ) derived from the map xy → y. In this case statement (d) combined with the shift is closely related to Thompson’s characterization of characters (see [18] for references); cf. Tachikawa’s derivation [24] of Brauer’s characterization from [21]. Roquette’s approach has strongly influenced the treatment here. Kletzing obtained his analogue [16, p. 114] of the ring Ap for K = Q as the localization of R (= DR) with respect to a certain prime ideal; I have not attempted to pursue this interesting approach. Gluck’s intricate characterization of m-monomial characters makes use of a proposition [11, Proposition 3] for cyclotomic K that bears some resemblance to our results. 5. Explicit constants Sometimes it is desirable to have an explicit integer constant in an induction theorem. The replacement argument based on Section 2 cannot give this by itself but together with Theorem 5.1 it can. Notations from previous sections will continue. The next result will let us control coefficient rings. Lemma 5.1. For any G, K, and g, g K = g F where F = Q(ωo(g) ) ∩ K.
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Proof. Without loss of generality we can suppose that G = g and then that h = o(g) in the definition of g K . Adjoining ωh to the terms of the relation Q ⊆ F ⊆ Q(ωh ) gives F (ωh ) = Q(ωh ); then F (ωh ) ∩ K = F . Now there is an isomorphism of Gal(K(ωh )/K) onto Gal(F (ωh )/F ), called inflation in [4, V, § 10, Theorem 5]; the actions of these Galois groups on ωh give the result. Lemma 5.2. Given G, K, and any g ∈ G; let Dg = Z[ωo(g) ] ∩ K. Then g K |NG (g K )|εgG,K ∈ IndG . g D R(Kg) | g Proof. This, essentially a known result, comes from Lemmas 3.2 and 3.3(d) for p = 0. (Of course g K = g g,K .) S]. Theorem 5.1. For any K-p -section S of G, |N |ε ∈ Ind[DR| Proof. Applying Lemma 5.1 to x we find that xK = xF with F = Q(ωo(x) ) ∩ K, whence S is also the F -p -section xG,F,p . Applying our usual notations to F instead of K we and D = Z[ωo(x) ] ∩ F = Z[ωo(x) ] ∩ K unchanged. Now ε = εsG,F can have x, N , P , G, summed over certain s ∈ S. By Lemma 4.1 we can choose each s within S and then with sp = x. In Lemma 5.2 for F and s, Ds = Z[ωo(s) ] ∩ F = D; by that lemma and the transitivity S] S] ⊆ Ind[DR| = W since NG (sF ) ⊆ NG (xF ) = of induction, |N |εsG,F ∈ Ind[DR(F G)| N. Theorem 5.1 strengthens a result of Fan and Hu [9, Proposition 2.7]. Theorems 4.1 and 5.1 together imply: Theorem 5.2. For any K-p -section S of G, S] ⊆ [DR|S] ∩ DV ⊆ DR. |N |p ε ∈ W = Ind[DR| = [DR| S]. o(x) For G, ε∈A For p = 0 this reduces to Lemma 5.2. 6. K-π -sections This section uses previous results to obtain corresponding facts involving sets of primes instead of p. For any set τ of primes, let τ be the complementary set and let nτ be the τ -part of any positive integer n. Define K-τ -sections as expected [9, p. 366], [13, 2.4.A]. Let π(G) be the set of prime divisors of |G|. I sometimes write p for {p}. Theorem 6.1. For any G and K and any nonempty set π of primes, let x be a π -element and T = xG,K,π the corresponding K-π -section. For any τ let DT (τ ) = Z[ωe(T )τ ] ∩ K
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where e(T ) is the least common multiple of the orders of the elements of T ; in particular DT (π) = Z[ωo(x) ] ∩ K. For each p ∈ π let VT (p) =
IndG H R(KH)
H
summed over all K-p-elementary subgroups H of G whose p -cyclic part has a generator t with π -part tπ equal to x [and such that the p-part of H is Sylow in NG (tK )]; let VT (π) = q∈π VT (q). Define WT (p) =
H,K,π IndG ] H [DT (π)R(KH)|x
H
for the same subgroups H and let WT (π) =
q∈π
WT (q). Then
(a) |NG (xK )|π εT ∈ DT (π)VT (π) ⊆ DT (π)R; (b) |NG (xK )|2π εT ∈ WT (π). Proof. Subscripts T will be omitted and dependence of t and other things on p and H will not be indicated. The bracketed clause is optional. Since primes outside π(G) have no effect on the assumption about T or on D(π) and the same effect (in fact no effect) on V(π) and W(π), we can assume that π is finite. Let p ∈ π. For each K-p -section S := tG,K,p ⊆ T , Theorem 5.2 gives a relation for |NG (tK )|p εS ; we can choose t ∈ S so that tπ = x. Combining these relations, v(p)ε ∈
Ind[D(p)R(KH)|tH,K,p ] ⊆
H
Ind[D(p)R(KH)|xH,K,π ]
(6.1)
H
where v(p) = |NG (xK )|p , with H as above. This implies that v(p)ε ∈
Ind D(p)R(KH) = D(p)V(p).
(6.2)
H
The greatest common divisor d = |NG (xK )|π of the constants v(p) is a Z-combination of them, whence dε ∈
D(p)V(p) ⊆ D(0)V(π).
(6.3)
p∈π
For m = |NG (xK )|, mε ∈ D(p)V(p) ⊆ D(p)V(π). Hence mε ∈
p∈π
D(p)V(π) =
p∈π
D(p) V(π) = D(π)V(π)
(6.4)
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228
where the first equality holds since V(π), as a Z-submodule of R, has a K-independent Z-basis (see the paragraph preceding (3.1) and [8, (4.14)]) and the second by intersections of cyclotomic fields [15, pp. 56–57], [7, Theorem 3.4], [17, Theorem 4.10(v)]. By (6.4), dε ∈ QD(π)V(π) ⊆ F (π)V(π) where F (π) is the field of quotients of D(π). Arguing as for (6.4) and using the fact that D(π) is the set of all algebraic integers in F (π), we have dε ∈ D(0)V(π) ∩ F (π)V(π) = D(0) ∩ F (π) V(π) = D(π)V(π), which is (a). = R(KH), T = xH,K,π , and ε = ε Writing R T T ,H for each H, we have [D(p)R|T ] = [D(p)R|T ]εT ⊆ D(p)RεT . With this (6.1) gives v(p)ε ∈ D(p)Z(p) where . Setting Z(π) = Z(p) = Ind Rε Z(q) and replacing V by Z in the three H
q∈π
T
paragraphs preceding this one, we find that dε ∈ D(π)Z(π). We can apply (a) to each (K-elementary) group H for G and its T for T to find = D(π)R, whence dε ∈ [D(π)R| T]. This implies that that |NH (xK )|π εT ∈ DT (π)R T ⊆ D(π)R[D(π) T] = [D(π)R| T], which gives dD(π)Z(π) ⊆ W(π). Since dD(π)Rε R| T 2 d ε = d(dε) ∈ dD(π)Z(π), we have (b). For π = {p}, (a) implies Theorem 3.1 but (b) does not quite imply Theorem 5.2 since the constants do not match. Theorem 6.1(a) generalizes a result (part of [9, Theorem 3.3]) that Fan and Hu used in their work on spectra, since it weakens the hypotheses by removing the “coprimely splitting” assumption. It also implies some results of Robinson [19, Lemma 3(i),(ii)] and of Robinson and Sin [20, Lemma]. 7. The common generalization Finally I can prove the common generalization mentioned in the title. Theorem 7.1. For any finite group G and field K of characteristic zero and any nonempty set π of primes, let VG (π) =
IndG H R(KH)
p∈π H
summed over all K-p-elementary subgroups H of G. Then |G|π 1G ∈ VG (π). Proof. By (6.3), |G|π 1G ∈ T DT (0)VT (π) ⊆ DG (0)VG (π) where DG (0) = Z[ωe(G)] ∩ K. Since this involves 1G , a well-known argument [3, Lemma 1], [8, (21.7)], [22, §12.7, Lemma 13] for replacing a domain by Z can be applied to yield the result.
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229
Theorem 7.2. For any G and K, if E is any integral domain of characteristic zero and if π is the set of primes (including 0) without inverses in E, 1G ∈ EVG (π). In particular, 1G ∈ Zπ VG (π) where Zπ =
p∈π
Zp .
This is a special case of the preceding theorem since |G|π is invertible in E. In the case that K = C this result is essentially due to Chen and Fan [6, Theorem 2.3]; as there, standard arguments give a characterization and induction theorem for EVG(π). The case in [6] is referred to in [5]. Theorem 7.2 is a true common generalization of the Witt-Berman theorem [2] (for E = Z) and a version of Artin’s induction theorem [22, §12.5, Theorem 26] (for E = Z0 = Q) since the definitions are the same for p = 0 as for other primes. (The proof for π = 0 does require some changes.) The wording requires that 0 ∈ π, but 0 can be omitted unless π = {0}. Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] B. Banaschewski, On the character rings of finite groups, Canad. J. Math. 15 (1963) 605–612. [2] S.D. Berman, Characters of linear representations of finite groups on an arbitrary field, translation by Morris D. Friedman, Inc., available at https://archive.org/details/nasa_techdoc_19880069124. (Accessed 26 February 2018). [3] R. Brauer, J. Tate, On the characters of finite groups, Ann. of Math. 62 (1955) 1–7. [4] N. Bourbaki, Algebra II, Chapters 4–7, Springer-Verlag, Berlin, 1990. [5] G. Chen, A generalized Brauer theorem and its converse, Algebra Colloq. 19 (2012) 427–432. [6] G. Chen, Y. Fan, On the connected components of the spectrum of the extended character ring of a finite group, J. Algebra 312 (2007) 689–698. [7] K. Conrad, Cyclotomic extensions, available at www.math.uconn.edu/~kconrad/blurbs/ galoistheory/cyclotomic.pdf. (Accessed 26 February 2018). [8] C.W. Curtis, I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. I, John Wiley & Sons, New York, 1981. [9] Y. Fan, X. Hu, On the connected components of the spectrums of the character rings of a finite group, J. Algebra 328 (2011) 355–371. [10] W. Feit, Characters of Finite Groups, W. A. Benjamin, Inc., New York, 1967. [11] D. Gluck, A characterization of generalized m-monomial characters, J. Algebra 71 (1981) 123–131. [12] D.M. Goldschmidt, I.M. Isaacs, Schur indices in finite groups, J. Algebra 33 (1975) 191–199. [13] K. Iizuka, Some studies on the orthogonality relations for group characters, Kumamoto J. Sci. Ser. A 5 (1961) 111–118. [14] I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. [15] G.J. Janusz, Algebraic Number Fields, second edition, American Mathematical Society, Providence, 1996. [16] D. Kletzing, Structure and Representations of Q-Groups, Lecture Notes in Mathematics, vol. 1084, Springer-Verlag, Berlin, 1984. [17] W. Narkiewicz, Elementary and Analytic Theory of Functions, second edition, SpringerVerlag/PWN–Polish Scientific Publishers, Berlin/Warsaw, 1990.
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[18] W.F. Reynolds, Thompson’s characterization of characters and sets of primes, J. Algebra 156 (1993) 237–243. [19] G.R. Robinson, Blocks, isometries, and sets of primes, Proc. Lond. Math. Soc. (3) 51 (1985) 432–448. [20] G.R. Robinson, P. Sin, A note on Brauer’s induction theorem, J. Algebra 162 (1993) 92–94. [21] P. Roquette, Arithmetische Untersuchung des Charakterringes einer endlichen Gruppe, J. Reine Angew. Math. 190 (1952) 148–168. [22] J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, New York, 1977. [23] L. Solomon, The representation of finite groups in algebraic number fields, J. Math. Soc. Japan 13 (1961) 144–164. [24] H. Tachikawa, A remark on generalized characters of groups, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 4 (1954) 332–334.
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Induction theorems for finite groups including a common generalization of two classical theorems William F. Reynolds 1 Department of Mathematics, Tufts University, Medford, MA 02155, USA
a r t i c l e
i n f o
Article history: Received 28 February 2018 Available online 16 March 2019 Communicated by Leonard L. Scott, Jr. MSC: 20C15 Keywords: Induction theorem Character ring K-elementary group p -section π -section Ring of functions Idempotent
a b s t r a c t Roquette proved results involving induction about rings related to the character rings of finite groups and consisting of functions that vanish outside a p -section; Kletzing did something similar for Q-characters. This paper gives new generalizations of these results involving K-characters for any field K of characteristic 0 as well as corresponding theorems for a set of primes. That these facts are useful is illustrated by showing that they imply a common generalization of the Witt-Berman and Artin induction theorems as well as results of Chen, Fan, Hu, Robinson, and Sin. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The best-known proof of Brauer’s induction theorem is probably that of Brauer and Tate [3]. Extensions of this proof to the Witt-Berman generalization over a field K of characteristic zero are found in [8, (21.6)] and [22, §12.6, Theorem 27 and Proposition 36]. These proofs use results about certain functions on a group G that vanish outside
1
E-mail address: fi[email protected]. Current address: 3 Preble Gardens Road, Belmont, MA 02478, USA.
https://doi.org/10.1016/j.jalgebra.2019.03.009 0021-8693/© 2019 Elsevier Inc. All rights reserved.
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a K-p -section of G for a prime p; this set of functions depends on a coefficient domain D. (Some terms are defined later.) In the case that K = C and D is p-adically complete for some p dividing p, Roquette extracted such results from earlier proofs and expanded them; in this way he reduced to a situation involving K-p-elementary subgroups (p-elementary in his case). Kletzing proved a similar result with K = Q, not assuming completeness. The aim of this paper is to present a generalization of Roquette’s and Kletzing’s reductions for arbitrary K, then a corresponding result for sets π of primes, and to demonstrate the power of these results by presenting a common generalization of the Witt-Berman and Artin induction theorems, thus strengthening a result of Chen and Fan that deals with the case that K = C. More specifically, let x be a p-regular element of G, let D = Z[ωo(x) ]∩K and Dp = Zp D where Zp consists of the p-integers in Q, and let R be the algebra of K-characters of G. A preliminary result, Theorem 3.1, states in effect that the characteristic function ε of the K-p -section S of x, an idempotent, is in Dp R. Thus Ap = (Dp R)ε is a direct summand—in fact indecomposable—of Dp R. Furthermore ε is induced from an element where R is the analogue of R for a K-p-elementary subgroup G = xP with P of Dp R as large as possible. Theorem 4.1 improves the induction statement and leads to Theorem 4.2, a reduction to K-p-elementary subgroups which is something like a Witt-Berman theorem on K-p -sections. This says, among other things, that Ap is induced bijectively p of Ap for G and that there is a restrictionfrom a certain subalgebra of the analogue A like isomorphism in the opposite direction. For K = C a connection between Roquette’s results and Thompson’s characterization of characters is pointed out. In Section 6, p is replaced by a set of primes and an induction theorem involving K-π -sections is proved. The common generalization is in the final section. My theorems also generalize or imply some results of Fan, Hu, Robinson, and Sin; this is evidence that they have other uses although they are rather technical. The proofs, like classical proofs of induction theorems, follow a pattern of first constructing a suitable function by induction and then replacing it by a constant times an idempotent. I have chosen to base the arguments for replacement on modifications of a lemma of Banaschewski, Goldschmidt, and Isaacs about functions on finite sets; these modifications are given in Section 2. By applying these instead of a well-known argument involving powers of functions, the intuitiveness of the proofs is arguably improved at the cost of using more standard facts from algebraic number theory. I never assume any form of Brauer’s theorem. 2. Functions on a finite set The lemmas in this section were developed from a lemma [12, Lemma 3.2], [14, Lemma 8.5], [8, Lemma 15.11] that can be stated in contrapositive form as follows: Lemma 2.1. (Banaschewski-Goldschmidt-Isaacs) Suppose that C is a “rng” (ring without identity assumed, or Z-algebra) of functions from a nonempty finite set T to Z such that
W.F. Reynolds / Journal of Algebra 528 (2019) 217–230
219
for every t ∈ T and every prime number p there exists ψ ∈ C such that p ψ(t). Then the 1-function 1T is in C. This follows easily from a result of Banaschewski [1, Proposition 1] on maximal ideals of rings of functions. (Hint: choose the ring in [1] to be generated by 1T and the “rng” in [12]). Goldschmidt and Isaacs attributed this result to Banaschewski but I think that it is different enough from his result to bear all three names, especially because it has a simple direct proof. Lemma 2.1 can be strengthened in three ways and then weakened (for simplicity) in one way to yield the next result. Let T be a nonempty finite set and R a commutative ring (with identity 1R ). For any ψ in the R-algebra F(T, R) of functions from T to R and any ideal p of R, let ε be the characteristic function of Sψ,p = {t ∈ T | ψ(t) ∈ / p}
(2.1)
in F(T, R). Sψ,p may be thought of as the support of ψ mod p. (Gluck [11, p. 126] has also used a modification of Lemma 2.1.) Lemma 2.2. For T and R as above, let ψ ∈ C where C is an R-subalgebra (no identity required) of F(T, R), i.e., an R-submodule that is closed under multiplication. Let p be a prime ideal of R. (The zero-ideal is allowed if it is prime, namely if R is an integral domain.) Then for ε as above, rε ∈ C for some r ∈ R, r ∈ / p. If Sψ,p is the same set for all p in a set Π of prime ideals, then the same r can be taken for all these ideals, so that r∈ / p for all p ∈ Π. Proof. Let S = Sψ,p and S = T \ S. First a “clearing” step, showing that we can replace ψ by some θ ∈ C with Sθ,p = Sψ,p such that θ vanishes on S . Taking S = ∅, define θ = ψ· s ∈S ψ−ψ(s )1T ; if we multiply θ out and look at the terms, we see that θ ∈ C; the factored form of θ(t), t ∈ T , shows that Sθ,p = Sφ,p since p is prime; the vanishing is obvious. Now expanding the equation s∈S θ(s)ε − θ = 0 proves the lemma since r = θ(s) ∈ / p; this is a “leveling” step replacing θ by rε. (If S = ∅, then θ = 0 = ε so that r = 1 works.) The last sentence of the lemma is true since the construction of r is independent of p for p ∈ Π. For use with some results such as Lemma 5.2 and Theorem 7.2, whenever I use the word “prime” as a noun I shall allow the possibility that the prime is 0; cf. [16, pp. 115, 140]. (Alternatively, to avoid unconventional terminology, I could use prime ideals of Z, including 0Z, instead of primes.) Definitions can be written so that they do not treat 0 individually, although the proofs may; I omit details. I shall not always give proofs for p = 0. Some algebraic number theory can give a constant in Z:
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Lemma 2.3. Given T , assume that R is the set D of all the algebraic integers in an algebraic number field F . Let ψ and C be as in Lemma 2.2 with ψ ∈ C and let p be a prime. Suppose that Sψ,p is the same set S for all p in the set Π of the prime ideals of D that contain p, except that if p = 0 suppose that S = Sψ,0D . Then kε ∈ C for some positive integer k, p k, where ε is the characteristic function of S on T . / p whenever p ∈ p. Proof. If p = 0, then Lemma 2.2 gives d ∈ D such that dε ∈ C while d ∈ We can replace d by a positive integer as follows. D is a Dedekind domain [8, pp. 76–78], [15, Theorem 3.16]. Hence if n is the norm of d from F to Q, then straightforwardly n ∈ Z, d = 0, n/d ∈ D, and p n (see [15, pp. 18, 20, 24, 42–44]); take k = ±n. Change this slightly if p = 0. (It is probably well known that [3, Theorem 3] can be proved by applying Lemma 2.1 to η there. Isaacs [14] used the same lemma differently in proving Brauer’s theorem.) 3. Induction from a K-p-elementary subgroup G will always be a finite group and K a field of characteristic 0. For any n let ωn be a primitive n-th root of unity in some extension of K. Let h be any multiple of the exponent e(G) of G and let L = K(ωh ). Then there is an action of the Galois group Gal(L/K) on G where each element acts on every cyclic subgroup g of G in the same way that it acts on ωo(g) , where o(g) is the order of g. The orbit g K of g (hence everything we care about) is independent of the choice of h; thus we can use the same L for subgroups of G. Combining this action with conjugation gives an action of G × Gal(L/K) on G, whose orbits are the K-conjugacy classes or K-classes g G,K . For any p -element (p-regular element) x of G, let xG,K,p be the K-p -section of x in G, i.e., the set of all elements g of G whose p -part gp is in xG,K ; similarly for subgroups. For any prime p I shall say that a group H is K-p-elementary or K-elementary for p if H = xP where x is a p -element and P a p-group that normalizes xK [8, p. 492], [22, §12.6]. Here the prime 0 behaves like any prime that does not divide |H| (or e(H)); in particular, “K-0-elementary” is the same as “cyclic.” Results for p = 0 are often trivial, but not always. Let R(KG) be the ring of K-characters of G and similarly for other groups and fields; I abbreviate R(KG) by R. For any integral subdomain D of K, the irreducible K-characters of G form a D-basis of the D-algebra DR (see [8, pp. 207, 493]), [22, §12.1, Proposition 32]) and a K-basis of the K-algebra of K-valued K-class-functions [8, (21.3)], [22, §12.4, Corollary 2]; hence the latter is KR. Thus DR can be identified with D ⊗Z R, etc. For any algebra J of functions with values in an extension of D, let the set of all functions in J with values in D be called JD . (This notion will be used to apply Lemmas 2.2 and 2.3.) For any union U of K-conjugacy classes of G, let εU = εU,G be the characteristic function of U on G, an idempotent, and similarly for other groups; thus εU ∈ KR.
(3.1)
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The following notation will be used throughout. Notation. Given any G, K, p, and a K-p -section S of G, choose a p -element x of G such that S = xG,K,p . Let N be the normalizer NG (xK ) in G of the K-class xK . Choose = xP , so that G is K-p-elementary. (The any p-Sylow subgroup P of N and let G choices of x and P do not really matter.) Let D = Z[ωo(x) ] ∩ K.
(3.2)
D consists of all the algebraic integers in Q(ωo(x) ) ∩ K [15, Theorem 10.4], [8, (4.5)]. Of course D depends on the section. Define ε = εS,G , ε = εS, G . Denote restriction and induction between G and G by Res and Ind without the usual indices and let S = xG,K,p , (⊆ R). = R(K G), V = Ind R R By the equation (Ind α)β = Ind(α Res β)
(3.3)
[8, (10.20)], [22, §7.2, Remark (3)], V is an ideal in the ring R and hence a Z-algebra. Most of my results carry over for suitable larger choices of D; the arguments throughout the paper would be somewhat simpler for Z[ωe(G) ] or Z[ω|G| ] as in references above but lead to weaker theorems. I have chosen domains that yield what seem to be natural forms of the results. The D of (3.2) can also be useful in special cases, e.g., if K ⊆ R. The proof of Theorem 3.1 follows the pattern described in the introduction: first I shall carry out an induction argument by modifying lemmas from some proofs of the Witt-Berman theorem, taking [8] and [22] as primary references (similar arguments can be found in [10] and [23], say); then I shall apply Lemma 2.3 to make a replacement argument. Lemma 3.1. Let H = xP be K-p-elementary. Then every K-character (i.e., trace of a representation over K) of x is the restriction of a K-character of H with values in Z[ωo(x) ] ∩ K. Proof. The restriction statement is in [8, (21.8)] and [22, §12.7, Lemma 17]. The value statement can be read off from [23, Lemma 2]. D such that Lemma 3.2. For any K-p -section S of G, there exists ξ ∈ (DR)
ResG x ξ = o(x) εxK ,x .
(3.4)
each with Proof. This is the same as [8, (21.9)] for G and [22, §12.7, Lemma 18] for G, minor adjustments and use of Lemma 3.1. (If K = C, as for Brauer’s theorem, then = x × P and we can replace D by Z in this lemma and below (see [3, §5, I]).) G
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D is such that ResG ξ = Lemma 3.3. For any K-p -section S of G, suppose that ξ ∈ (DR) x aεxK ,x for some positive integer a with p a. Let ψ = Ind ξ. Then (a) (b) (c) (d)
ψ ∈ (DV)D ; ψ(g) = 0 if g is a p -element of G not in xG,K ; ≡ 0 (mod p); ψ(x) = a|N : G| if also a = o(x) as in (3.4), then ψ(x) is the p -part of |N |.
Proof. This follows from the proof of [8, (21.11)] or [22, §12.7, Lemma 18]. Lemma 3.4. For any G, K, and p, let E be an integral domain included in Z[ωe(G) ] and suppose that p is a prime ideal of E such that p ∈ p. If θ ∈ (ER)E (the superscript being like D above) and if g and g are in the same K-p -section of G, then θ(g) ≡ θ(g ) (mod p). Proof. Apply [8, (21.12)] to get the corresponding result for Z[ω|G| ] or use [22, §12.7, Lemma 16] similarly for Z[ωe(G) ]; then reduce to E using [15, pp. 28–29]. Lemma 3.5. Under the assumptions of Lemma 3.3, ψ(g) ≡ a|N : G|ε(g)
(mod p)
(3.5)
for all g ∈ G and for all prime ideals p of D such that p ∈ p (recall that ε = εS,G ). Then Sψ,p = S for all such p if p = 0 and for p = 0D if p = 0. If also ψ has values in Z, then Sψ,pZ = S. Proof. (3.5) is immediate from the preceding two lemmas; the function ψ corresponds to ψj in [8, p. 498, Step 1] and to ψλ in [22, §12.7, Lemma 19]. By (2.1) the other statements follow since ψ(x) ∈ / pZ = p ∩ Z. Finally Lemma 2.3 comes into play to give a result involving the K-p-elementary group G. Theorem 3.1. For any K-p -section S of G, ⊆ DR kε ∈ DV = Ind DR = DR(K G) with D as in (3.2). for some positive integer k, p k, where DR Proof. In the hypothesis of Lemma 2.3, take T = G, D and p as given, ψ = Ind ξ with ξ given by Lemma 3.2, and C = (DV)D , a D-algebra. By Lemma 3.3(a), ψ ∈ C. Then Sψ,p = S for p as in Lemma 3.5; hence Lemma 2.3 applies and gives the result. (For p = 0 this theorem can be obtained more directly from Lemma 5.2 below.)
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As in the introduction, Theorem 3.1 can be stated in terms of a semilocal integral domain; see the next section. It will be improved by Theorems 4.1 and 5.2. 4. Induction from a K-p -section In this section we make the set of inducing functions in Theorem 3.1 smaller and then localize the coefficient domain to obtain Theorem 4.2. Lemma 4.1. For any K-p -section S of G, every element of S is conjugate in G to an ˜ element of S. Proof. (Cf. [21, (26)].) Replacing s ∈ S by a conjugate in G, we can suppose that sp ∈ xK . Then the p-part of s is in some p-Sylow subgroup Q1 of C := CG (x) and s ∈ xK Q1 . We can replace the new s by a conjugate in N (= NG (xK )) to get a third and sp ∈ xK s ∈ xK Q with Q included in the p-Sylow subgroup P of N , whence s ∈ G (It follows that Q = P ∩ C.) so that s ∈ S. For any algebra J of K-class-functions on a group G and any subset T of G, let [J|T ] consist of all those functions in J that vanish outside T . Let S be a K-p -section of G. By (3.1), ε ∈ KR; hence F := (KR)ε = [KR|S] is a := (K R) ε= ring with identity ε and also a direct ideal summand of KR; similarly for F [K R|S]. For φ ∈ F define ResSS φ = (Res φ) ε ∈ F.
(4.1)
This modification of restriction is similar to [21, (29)] and [16, pp. 145-146]. By a statement preceding (3.1), F is the set of all K-class-functions from G to K that vanish outside S, with {εsG,K | s ∈ S} as a K-basis. ResSS εsG,K = εsG,K ∩S , which is nonzero by Lemma 4.1. Thus a K-basis of ResSS F is given by the distinct characteristic such that θ has G denote the set of functions θ ∈ F Let F functions εtG,K ∩S , t ∈ S. the same value on any two elements of S that are K-conjugate in G, with corresponding notations for similar situations. (Gluck [11, p. 125] calls functions like these G-invariant.) N , Ind gives a K-space-isomorphism G = F Lemma 4.2. For any K-p -section S of G, F S N to F, and Res is a K-algebra-isomorphism of F to F N . of F S G . F N since a conjugating element fixes S and hence G = F Proof. Clearly ResSS F = F
fixes xG,K = xK . The isomorphisms follow from the above description of K-bases, since the formula for induction implies that Ind εtG,K ∩S˜ is a positive multiple of εtG,K . Now I shall build on Theorem 3.1 by repeating the induce-replace pattern; this time the output of that theorem provides the input of the induction argument. Thus the entire proof of Theorem 4.1 requires going through the pattern twice.
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Define D-algebras A := DR ∩ F = [DR|S],
:= DR ∩F = [DR| S], A
N := A ∩F N . A
(There is an analogy of A to the algebra of virtual or generalized principal indecomposable characters and of B below to the Brauer character algebra: see, for example, [18, p. 239].) By Theorem 3.1 there exist positive integers k and l, not divisible by p, such that in fact, l N . Then kε ∈ DR whence kε ∈ A and similarly l ε ∈ A; ε∈A k ε = ResSS kε ∈ ResSS A =: Y,
k Ind ε ∈ Ind Y =: X.
(4.2)
Y is a D-algebra since ResSS preserves products. Consider the D-algebra B := (DR)ε; clearly A ⊆ B. If β ∈ B, then β = θε, θ ∈ DR. A ⊆ A; hence we can write lY ⊆ Then ResSS lβ = (Res θ)(Res lε) ε = (Res θ)(l ε) ⊆ (DR) S N . By induction lX ⊆ Ind A N ⊆ Ind A := W. Then ResS lB ⊆ A lX ⊆ W ⊆ A ⊆ B.
(4.3)
If γ ∈ X and δ ∈ A, then γ = Ind ζ, ζ ∈ Y, and ζ Res δ = ζ ε Res δ = ζ ResSS δ. By (3.3), γδ = Ind(ζ ResSS δ) ∈ Ind YY ⊆ Ind Y = X so that XA ⊆ X. By (4.3) lX ⊆ A, so that lXX ⊆ XA ⊆ X and lX is a D-algebra. Now to use Lemma 2.2: take T = G, R = D = Z, p = pZ, ψ = kl Ind ε, and C = (lX)Z . and x ∩ S = x ∩ xG.K Since l ε∈A = x ∩ xK = xK , ξ = kl ε satisfies the hypothesis of Lemma 3.3. Since ψ = Ind ξ, Lemma 3.5 gives Sψ,pZ = S. By (4.2) ψ ∈ lX, hence ψ ∈ C. Lemma 2.2 now implies that vε ∈ lX ⊆ X with p v; together with (4.3) this implies: Theorem 4.1. For any K-p -section S of G, S] ⊆ [DR|S] ∩ DV ⊆ DR vε ∈ X = Ind ResSS A and vε ∈ W = Ind[DR|
(4.4)
for some positive integer v not divisible by p. The second statement strengthens Theorem 3.1; in turn it will be made explicit by Theorem 5.2. If α ∈ A, then vα = vεα ∈ XA ⊆ X; thus vA ⊆ X.
(4.5)
N ⊆ vW ⊆ X = Ind Y and Lemma 4.2 implies that v A N ⊆ Y. Hence Ind v A To formulate the next theorem I introduce the semilocal integral domain Dp := Zp D as coefficient ring. (I have postponed doing this in order to get information about constants in Z in the next section.) For any of our D-modules M, let Mp = Zp M, a Dp -module. N )p = A p ∩ F N , Bp = (Dp R)ε, etc. By (4.3) and (4.5) Easily, Ap = Zp A = [Dp R|S], (A
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p = Ap . Since lY ⊆ A N and v A N ⊆ Y, we have Xp = Wp = Ap , i.e., Ind Yp = Ind A Yp = (AN )p . We now have the following reduction to a situation involving a K-p-elementary sub N )p = A p ∩ F N and similarly with G in place of N , this theorem can group. Since (A be stated entirely in terms of Dp -modules. A corresponding reduction in which Dp is replaced by a larger local ring Dp can be deduced easily from it. Theorem 4.2. For any K-p -section S of G, (a) ε ∈ Ap and Ap = [Dp R|S] = (Dp R)ε is an indecomposable direct summand of Dp R = p; Dp R(KG); similarly for A N )p to Ap ; (b) Ind(AN )p = Ind Ap = Ap and Ind gives a Dp -module-isomorphism of (A S S N )p ; (c) ResS Ap = (AN )p = (AG )p and ResS gives a Dp -algebra-isomorphism of Ap to (A N )p , and ResS φ ∈ A p are equivalent. (d) if φ ∈ F, then the relations φ ∈ Ap , ResSS φ ∈ (A S Proof. This is mostly a summation of the results above. In particular by Theorem 4.1, ε ∈ Xp = Ap whence Ap = Bp . The idempotent ε of Dp R is indecomposable: a proper summand of ε would have values 0 and 1; lifting it to DR and applying Lemma 3.4 gives a contradiction. Hence Ap is an indecomposable direct summand. For K = C Roquette [21, §3] proved results equivalent to almost all of this theorem with the p-adic completion of Z[ωe(G) ] instead of Dp . He took advantage of the fact that = x × P to use functions on P instead of on S = xP ; if K = C we could do he had G p = [Dp R(C G)|xP the same, shifting by the isomorphism of A ] to Dp R(CP ) derived from the map xy → y. In this case statement (d) combined with the shift is closely related to Thompson’s characterization of characters (see [18] for references); cf. Tachikawa’s derivation [24] of Brauer’s characterization from [21]. Roquette’s approach has strongly influenced the treatment here. Kletzing obtained his analogue [16, p. 114] of the ring Ap for K = Q as the localization of R (= DR) with respect to a certain prime ideal; I have not attempted to pursue this interesting approach. Gluck’s intricate characterization of m-monomial characters makes use of a proposition [11, Proposition 3] for cyclotomic K that bears some resemblance to our results. 5. Explicit constants Sometimes it is desirable to have an explicit integer constant in an induction theorem. The replacement argument based on Section 2 cannot give this by itself but together with Theorem 5.1 it can. Notations from previous sections will continue. The next result will let us control coefficient rings. Lemma 5.1. For any G, K, and g, g K = g F where F = Q(ωo(g) ) ∩ K.
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Proof. Without loss of generality we can suppose that G = g and then that h = o(g) in the definition of g K . Adjoining ωh to the terms of the relation Q ⊆ F ⊆ Q(ωh ) gives F (ωh ) = Q(ωh ); then F (ωh ) ∩ K = F . Now there is an isomorphism of Gal(K(ωh )/K) onto Gal(F (ωh )/F ), called inflation in [4, V, § 10, Theorem 5]; the actions of these Galois groups on ωh give the result. Lemma 5.2. Given G, K, and any g ∈ G; let Dg = Z[ωo(g) ] ∩ K. Then g K |NG (g K )|εgG,K ∈ IndG . g D R(Kg) | g Proof. This, essentially a known result, comes from Lemmas 3.2 and 3.3(d) for p = 0. (Of course g K = g g,K .) S]. Theorem 5.1. For any K-p -section S of G, |N |ε ∈ Ind[DR| Proof. Applying Lemma 5.1 to x we find that xK = xF with F = Q(ωo(x) ) ∩ K, whence S is also the F -p -section xG,F,p . Applying our usual notations to F instead of K we and D = Z[ωo(x) ] ∩ F = Z[ωo(x) ] ∩ K unchanged. Now ε = εsG,F can have x, N , P , G, summed over certain s ∈ S. By Lemma 4.1 we can choose each s within S and then with sp = x. In Lemma 5.2 for F and s, Ds = Z[ωo(s) ] ∩ F = D; by that lemma and the transitivity S] S] ⊆ Ind[DR| = W since NG (sF ) ⊆ NG (xF ) = of induction, |N |εsG,F ∈ Ind[DR(F G)| N. Theorem 5.1 strengthens a result of Fan and Hu [9, Proposition 2.7]. Theorems 4.1 and 5.1 together imply: Theorem 5.2. For any K-p -section S of G, S] ⊆ [DR|S] ∩ DV ⊆ DR. |N |p ε ∈ W = Ind[DR| = [DR| S]. o(x) For G, ε∈A For p = 0 this reduces to Lemma 5.2. 6. K-π -sections This section uses previous results to obtain corresponding facts involving sets of primes instead of p. For any set τ of primes, let τ be the complementary set and let nτ be the τ -part of any positive integer n. Define K-τ -sections as expected [9, p. 366], [13, 2.4.A]. Let π(G) be the set of prime divisors of |G|. I sometimes write p for {p}. Theorem 6.1. For any G and K and any nonempty set π of primes, let x be a π -element and T = xG,K,π the corresponding K-π -section. For any τ let DT (τ ) = Z[ωe(T )τ ] ∩ K
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where e(T ) is the least common multiple of the orders of the elements of T ; in particular DT (π) = Z[ωo(x) ] ∩ K. For each p ∈ π let VT (p) =
IndG H R(KH)
H
summed over all K-p-elementary subgroups H of G whose p -cyclic part has a generator t with π -part tπ equal to x [and such that the p-part of H is Sylow in NG (tK )]; let VT (π) = q∈π VT (q). Define WT (p) =
H,K,π IndG ] H [DT (π)R(KH)|x
H
for the same subgroups H and let WT (π) =
q∈π
WT (q). Then
(a) |NG (xK )|π εT ∈ DT (π)VT (π) ⊆ DT (π)R; (b) |NG (xK )|2π εT ∈ WT (π). Proof. Subscripts T will be omitted and dependence of t and other things on p and H will not be indicated. The bracketed clause is optional. Since primes outside π(G) have no effect on the assumption about T or on D(π) and the same effect (in fact no effect) on V(π) and W(π), we can assume that π is finite. Let p ∈ π. For each K-p -section S := tG,K,p ⊆ T , Theorem 5.2 gives a relation for |NG (tK )|p εS ; we can choose t ∈ S so that tπ = x. Combining these relations, v(p)ε ∈
Ind[D(p)R(KH)|tH,K,p ] ⊆
H
Ind[D(p)R(KH)|xH,K,π ]
(6.1)
H
where v(p) = |NG (xK )|p , with H as above. This implies that v(p)ε ∈
Ind D(p)R(KH) = D(p)V(p).
(6.2)
H
The greatest common divisor d = |NG (xK )|π of the constants v(p) is a Z-combination of them, whence dε ∈
D(p)V(p) ⊆ D(0)V(π).
(6.3)
p∈π
For m = |NG (xK )|, mε ∈ D(p)V(p) ⊆ D(p)V(π). Hence mε ∈
p∈π
D(p)V(π) =
p∈π
D(p) V(π) = D(π)V(π)
(6.4)
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where the first equality holds since V(π), as a Z-submodule of R, has a K-independent Z-basis (see the paragraph preceding (3.1) and [8, (4.14)]) and the second by intersections of cyclotomic fields [15, pp. 56–57], [7, Theorem 3.4], [17, Theorem 4.10(v)]. By (6.4), dε ∈ QD(π)V(π) ⊆ F (π)V(π) where F (π) is the field of quotients of D(π). Arguing as for (6.4) and using the fact that D(π) is the set of all algebraic integers in F (π), we have dε ∈ D(0)V(π) ∩ F (π)V(π) = D(0) ∩ F (π) V(π) = D(π)V(π), which is (a). = R(KH), T = xH,K,π , and ε = ε Writing R T T ,H for each H, we have [D(p)R|T ] = [D(p)R|T ]εT ⊆ D(p)RεT . With this (6.1) gives v(p)ε ∈ D(p)Z(p) where . Setting Z(π) = Z(p) = Ind Rε Z(q) and replacing V by Z in the three H
q∈π
T
paragraphs preceding this one, we find that dε ∈ D(π)Z(π). We can apply (a) to each (K-elementary) group H for G and its T for T to find = D(π)R, whence dε ∈ [D(π)R| T]. This implies that that |NH (xK )|π εT ∈ DT (π)R T ⊆ D(π)R[D(π) T] = [D(π)R| T], which gives dD(π)Z(π) ⊆ W(π). Since dD(π)Rε R| T 2 d ε = d(dε) ∈ dD(π)Z(π), we have (b). For π = {p}, (a) implies Theorem 3.1 but (b) does not quite imply Theorem 5.2 since the constants do not match. Theorem 6.1(a) generalizes a result (part of [9, Theorem 3.3]) that Fan and Hu used in their work on spectra, since it weakens the hypotheses by removing the “coprimely splitting” assumption. It also implies some results of Robinson [19, Lemma 3(i),(ii)] and of Robinson and Sin [20, Lemma]. 7. The common generalization Finally I can prove the common generalization mentioned in the title. Theorem 7.1. For any finite group G and field K of characteristic zero and any nonempty set π of primes, let VG (π) =
IndG H R(KH)
p∈π H
summed over all K-p-elementary subgroups H of G. Then |G|π 1G ∈ VG (π). Proof. By (6.3), |G|π 1G ∈ T DT (0)VT (π) ⊆ DG (0)VG (π) where DG (0) = Z[ωe(G)] ∩ K. Since this involves 1G , a well-known argument [3, Lemma 1], [8, (21.7)], [22, §12.7, Lemma 13] for replacing a domain by Z can be applied to yield the result.
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Theorem 7.2. For any G and K, if E is any integral domain of characteristic zero and if π is the set of primes (including 0) without inverses in E, 1G ∈ EVG (π). In particular, 1G ∈ Zπ VG (π) where Zπ =
p∈π
Zp .
This is a special case of the preceding theorem since |G|π is invertible in E. In the case that K = C this result is essentially due to Chen and Fan [6, Theorem 2.3]; as there, standard arguments give a characterization and induction theorem for EVG(π). The case in [6] is referred to in [5]. Theorem 7.2 is a true common generalization of the Witt-Berman theorem [2] (for E = Z) and a version of Artin’s induction theorem [22, §12.5, Theorem 26] (for E = Z0 = Q) since the definitions are the same for p = 0 as for other primes. (The proof for π = 0 does require some changes.) The wording requires that 0 ∈ π, but 0 can be omitted unless π = {0}. Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] B. Banaschewski, On the character rings of finite groups, Canad. J. Math. 15 (1963) 605–612. [2] S.D. Berman, Characters of linear representations of finite groups on an arbitrary field, translation by Morris D. Friedman, Inc., available at https://archive.org/details/nasa_techdoc_19880069124. (Accessed 26 February 2018). [3] R. Brauer, J. Tate, On the characters of finite groups, Ann. of Math. 62 (1955) 1–7. [4] N. Bourbaki, Algebra II, Chapters 4–7, Springer-Verlag, Berlin, 1990. [5] G. Chen, A generalized Brauer theorem and its converse, Algebra Colloq. 19 (2012) 427–432. [6] G. Chen, Y. Fan, On the connected components of the spectrum of the extended character ring of a finite group, J. Algebra 312 (2007) 689–698. [7] K. Conrad, Cyclotomic extensions, available at www.math.uconn.edu/~kconrad/blurbs/ galoistheory/cyclotomic.pdf. (Accessed 26 February 2018). [8] C.W. Curtis, I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. I, John Wiley & Sons, New York, 1981. [9] Y. Fan, X. Hu, On the connected components of the spectrums of the character rings of a finite group, J. Algebra 328 (2011) 355–371. [10] W. Feit, Characters of Finite Groups, W. A. Benjamin, Inc., New York, 1967. [11] D. Gluck, A characterization of generalized m-monomial characters, J. Algebra 71 (1981) 123–131. [12] D.M. Goldschmidt, I.M. Isaacs, Schur indices in finite groups, J. Algebra 33 (1975) 191–199. [13] K. Iizuka, Some studies on the orthogonality relations for group characters, Kumamoto J. Sci. Ser. A 5 (1961) 111–118. [14] I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. [15] G.J. Janusz, Algebraic Number Fields, second edition, American Mathematical Society, Providence, 1996. [16] D. Kletzing, Structure and Representations of Q-Groups, Lecture Notes in Mathematics, vol. 1084, Springer-Verlag, Berlin, 1984. [17] W. Narkiewicz, Elementary and Analytic Theory of Functions, second edition, SpringerVerlag/PWN–Polish Scientific Publishers, Berlin/Warsaw, 1990.
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