On Integral Representations over Cyclotomic Fields

Journal of Number Theory  NT2007 journal of number theory 61, 4451 (1996) article no. 0136

On Integral Representations over Cyclotomic Fields G.-Martin Cram and Olaf Nei?e* Institut fur Mathematik der Universitat Augsburg, 86135 Augsburg, Germany Communicated by K. Rubin Received September 18, 1995

1. INTRODUCTION In 1945 Brauer [2] observed that every irreducible complex character / of a finite group G can be realized by a representation of G over the cyclotomic field Q(` e ), where e denotes the exponent of G and ` e is a primitive eth root of unity. Recently, Cliff et al. [1] proved for absolutely irreducible characters of solvable groups the existence of an integral representationa representation in matrices over Z[` e ]that realizes the character. Obviously the field Q(` e ) depends on the group only, not on the character. Ritter and Weiss suspected that given a group G of odd order and an absolutely irreducible character / of G there exists a representation by matrices that realizes / over Z[` f/ ], whereby f / denotes the conductor of the character field Q(/), i.e., f / :=min[n # N | Q(/)Q(` n )]. The main result of our article is the proof of this conjecture. More precisely, we prove the following: 1.1. Theorem. Let G be a finite solvable group and let / be an irreducible complex character of G with odd degree /(1). By f / we denote the conductor of the abelian field Q(/) and by ` f/ a primitive f / th root of unity. Then there exists a representation of G by matrices over Z[` f/ ] belonging to the character /. 1.2. Corollary. In the situation of Theorem 1.1 the Schur index s Q(`f/ )(/) of / over Q(` f/ ) is trivial. By definition of the conductor, Q(` f/ ) is the minimal cyclotomic field containing the character field. Assuming the conditions of the theorem, our theorem generalizes therefore the following results: v Solomon has shown in [10] (see [5, (10.15)]) that s K (/)=1, where K denotes the cyclotomic field Q(/, ` n ), n :=> p | |G| p. * The second author acknowledges financial support provided by the DFG.

44 0022-314X96 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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v If /(1) is an odd prime Jonas has proved in [6, Satz 2] that s Q(`f/ )(/)=1. v For primitive characters Theorem 1.1(a) in [8] of Opolka states that s Q(/)(/)=1. But in this case Q(/) is a cyclotomic field (see [7, Bemerkung 6.5]). v Since the character field of any irreducible character of a nilpotent group is a cyclotomic field, Theorem 1.1 was proved for nilpotent groups by Roquette [9] (see [5, (10.14)]). v A weaker form of the theorem was proved in [7]. As we will see, the proof of our theorem turns out to be constructive. In Part 4 we show by examples that in our theorem the odd degree condition for the characters is necessary.

2. PRELIMINARY RESULTS Suppose that G is a finite group and Irr(G ) denotes the set of absolutely irreducible complex characters. Given a character / # Irr(G) we ask whether there exists a representation that realizes /. First we try to construct the representation over K :=Q(` f/ ) by inducing representations from subgroups. The easy case occurs when there is a representation by matrices over K of a subgroup U
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g

(x) :=

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CRAM AND NEI;E

Proof. We can obviously assume Q(/)K. By Clifford theory, it holds for _ # G e}

: gU # GU

g

=res GN (/)=(res GN (/)) _ =e }

:

g

_

gU # GU

for some e # N. Each _ # G corresponds by this to a unique gU # GU with _ = g~ for g~ # gU. This defines a map } : G  GU, which is obviously injective. Since Stb G ( _ )=Stb G ()=U we know g~U=Ug~ for _ # G, g~ # }(_), and so U \V :=( }(_) | _ # G). It is easy to see that } : G  VU is multiplicative, i.e. an isomorphism. Furthermore by Clifford theory there exists a unique  # Irr(U ) with ind GU()=/ and res U N()=e. Therefore K()=K() and  _ corresponds to  _. For | :=ind VU () we know |(x)= gU # VU g= _ # G  _(x) if x # U, and |(x)=0 otherwise. Hence K(|)=K and ind GV (|)=/. Because of the K-primitivity of / we conclude V=G, which shows the assertion. K The above lemma yields for a K-primitive / # Irr(G ) that all characters in Irr(/ | N ), N \G, are Galois conjugate and therefore have the same kernels. In particular we have the 2.2. Corollary. If / # Irr(G ) is faithful and K-primitive for KC, then for N \G every irreducible constituent of res GN (/) is faithful as well. Especially every abelian normal subgroup of G is cyclic. We conclude this section with the proof of a special case of Theorem 1.1. 2.3. Proposition. Let G be a finite group, / # Irr(G ) faithful and Q(` f/ )-primitive. Suppose N is a normal abelian subgroup of G with C G (N )=N and odd index [G : N ]. Then / can be realized as a representation by matrices over Z[` f/ ]. Proof. From the Corollary 2.2 we know that N is cyclic, N=( x), and from the Lemma 2.1 we have GN& G :=Gal(L | K), where K :=Q(` f/ ),  # Irr(/ | N ) and L :=K(). Since K and L are cyclotomic fields and n :=[G : N ] is odd, all roots of unity of prime order in L lay in K. Therefore G must be cyclic as well, and the norm map N LK is surjective on the sets of roots of unity. We take a generator _ of G and choose g # G such that _ corresponds to gN by the isomorphism } of Lemma 2.1. Note that (g n ) is a root of unity lying in K because of the identity  _(g n )= g( g n )=(g n ). Hence there exists a root of unity ; # L with N LK ( ;)=( g n ). In addition we fix a root of unity ! # L such that L=Q(!). We regard L as a vectorspace over K with base [1, !, ! 2, ..., ! n&1 ]. Then the multiplication on L leads to a

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monomorphism =: L  Mat n(K ) with =(Z[!])Mat n(Z[` f/ ]). Since LK is an extension of cyclotomic fields we know ! _ =! a for some a # N, and if we define the matrix C # Gl n(Z[` f/ ]) by c ij :=

{

` n(a&1+( j&ai )n) 0

in the case n | ( j&ai ) otherwise,

then an elementary calculation shows us C &1=(:) C==(: _ ) for : # L. We set 2(x) :==((x)) and 2(g) :==(;) C. In fact this defines a representation by matrices 2 of G that realizes / over Z[` f/ ]. K For more details and a more general statement on this proposition see [7], Chapter 5.

3. PROOF OF THE THEOREM Our proof of Theorem 1.1 resembles to the main proof in [1]. Let G be a finite solvable group and take / # Irr(G ) of odd degree. We use an induction argument on the order of the group. Therefore we may assume that / is faithful, nonlinear and K-primitive, where K denotes Q(` f/ ). We choose a maximal abelian normal subgroup A IG. It follows from Corollary 2.2 that A is cyclic. For . # Irr(/ | A) Lemma 2.1 tells us that S :=Stb G (.)= C G (A) is normal in G and GS & Gal(K(.) | K ). In the case S=A we can use Proposition 2.3 which gives us the required explicit representation. Now we investigate the case S>A. Choose a normal subgroup N \G such that A
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Next we claim that E is normal in G. For this we prove E=0 1(P), which is a characteristic subgroup of P. Since E0 1(P), we get exp(E )=p. By construction every element x of P can be written as x=e } z with e # E and z # Z(P). Since (ez) p =e pz p =z p =1 iff z # 0 1(Z(P))=Z(E ), every element of order p of P is an element of E, which proves the claim. In particular this shows Z(E ) \G. But we will even obtain Z(E ) Z(G ): For * # Irr(/ | Z(E )) we know Q(*)=Q(` p ) and, by Lemma 2.1, | Gal(K(` p ) | K )| = | Gal(K(*) | K )| = | GStb G (*)| = | GC G (Z(E ))|, which divides /(1) by Clifford theory. Both fields K(` p ) and K are cyclotomic fields and, by the assumption on /(1) being odd, they must coincide. For that reason C G (Z(E ))=G

and

` p # K.

Now we are able to use character analysis as Dade describes it in [3, Sects. 5, 6]. To do this we have to deduce the facts of the following hypotheses (5.1) given in [3]: (5.1a)

E is a normal extra-special p-subgroup of G,

(5.1b)

G centralizes the center Z(E ) of E,

(5.1c) there is a normal subgroup L 0 of the factor group GC G (EZ(E )), (5.1d)

|L 0 | is not divisible by p,

(5.1e)

[EZ(E ), L]=EZ(E ).

Since NA = EAA & EE & A = EZ(E ) is a chief section of G, G operates irreducibly on EZ(E ). Hence G 0 :=GC G (EZ(E )) acts irreducibly and faithfully on EZ(E ). An easy calculation shows C := C G (EZ(E ))=EC G (E ). Let LC be a normal subgroup of G such that L 0 :=LC is a minimal normal subgroup of G 0 . For a picture see Fig. 1. Since G 0 is solvable, L 0 is an elementary abelian q-subgroup for a prime q and L 0 acts without fixpoints on the F p -vectorspaces EZ(E ) &CC G (E ), where F p denotes the finite field ZpZ. For that reason q{ p and by [4, 13.4b] EZ(E )=[EZ(E ), L 0 ]. The hypotheses (5.1) in [3] hold and applying (5.2) in [3] we receive a complement H of the section EZ(E ) satisfying HE=G, H & E=Z(E ) and C G (E )H. For the rest of the proof we regard the semidirect products G =E < H and G 0 :=E < H 0 , where H and H 0 :=HC G (E ) act on E by conjugation. Obviously G 0 is a factor group of G, and G is a factor group of G too by the epimorphism } : G  G, (e, h) [ eh. Each | # Irr(/ | E ) is faithful and takes values in Q(` p ). Using the fact ` p # K and Lemma 2.1 we conclude Stb G (|)=G, i.e. Irr(/ | E )=[|].

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The main idea is now to extend |, regarded as a character on E < 1G, to a character  on G that can be realized over Z[` p ], and to compare this with ``/''. The existence of the extension originate in works of Ward [11], Dade [3] and CliffRitterWeiss [1]. We summarize the results we need here in a theorem. 3.1. Theorem. Let G 0 =E < H 0 be the semidirect product of an extra-special p-group E ( p an odd prime) with a finite group H 0 such that the center of E is central in G 0 , and [E, K 0 ]=E for some normal subgroup K 0 of H 0 of order prime to p. Then any faithful irreducible character | of E extends to a character  of G 0 realizable as representation of matrices over Z[` p ]. For the proof we refer to Proposition 1.2 and Lemma 1.5 in [1]. We remark that the proofs there are constructive. In fact Z(E )Z(G ) implies Z(E )Z(G 0 ), and since for K :=H & L we know E=[E, L]=[E, K ], we can conclude E=[E, K 0 ] for the q-group K 0 :=KC G (E ) &LC=L 0 , regarded as a subgroup of H 0 in G 0 . We illustrate the situation in Fig. 1. So we can apply the Theorem 3.1 on G 0 =E < H 0 and receive a character  # Irr(G 0 ) extending |, which can be realized by a representation 9 of matrices over Z[` p ]. We inflate 9 to the representation 9 with character  # Irr(G ). Then we inflate / via } to the character /^ of G and notice that res GE (/^ ) is a multiple of |. By a result of Gallagher (see [5, Cor. (6.17)]) there exists a unique character  # Irr(GE ) such that /^ = , whereby  # Irr(G ) denotes the inflated character of  in G. Since GE &H
Figure 1

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Let T be such a realization of  over Z[` f ]. To finish the proof of Theorem 1.1 we observe that the uniqueness of  and the fact Q()= Q(` p )K implies Q()K, hence f  | f / , since any nontrivial Galois automorphism _ # Gal(K( ) | K ) would give another such character. Hence 9  T is a realization of / over Z[` f/ ] and Theorem 1.1 is proved. 3.2. Corollary. Every irreducible character / of a finite group of odd order can be realized by matrices over Z[` f/ ].

4. EXAMPLES Theorem 1.1 fails, even on field level, if we dismiss the odd degree hypothesis. For instance the nonlinear character / of the quaternion group takes values in Q, but has nontrivial Schur index s Q (/)=2. Usually, statements become true when one adjoines a 4th root of unity. We will now give an example which shows that for Theorem 1.1 the obstruction is independent of the 2-part of the conductor of the field. 4.1. Example. We take two odd primes p, q and a number t # N 0 such that p t | (q&1). Let A=( a) denote a cyclic group of order q and let B=( b) denote a cyclic group of order p t(q&1). Since Aut(A) is cyclic of order q&1, there exists an epimorphism # : B  Aut(A). Using this we define G :=A < B as the semidirect product of A with B. The elements a and b q&l generate a cyclic normal subgroup N of order p tq in G. We choose a faithful linear character { of N and observe that / :=ind G N({) is a faithful irreducible character of G with degree /(1)=q&1. Since B operates on A as the full Galois group Gal(Q(` q ) | Q) on ` q we receive Q(/)= Q(` pt )=Q(` f/ ). Now we claim s Q(/)(/)=p t. For details of the proof of this claim see e.g. [7, Beispiel 1.10]. First we remark that the Schur index coincides with the order of {(b q&1 ) N LK (L _ ) # K _N LK (L _ ), where K :=Q(/) and L :=Q({). Since ` :={(b q&1 ) is a root of unity of order p t, we know ord(`N LK (L _ )) | p t. For the other direction we consider the situation modulo the primes over q and take into account that LK is totally ramified over q. In particular, the Schur index is odd and will not change, if we increase the field by adjoining cyclotomic units of 2-power order. The condition on G being solvable is indispensable for our proof of Theorem 1.1. But up to date the authors don't know about an example of an odd degree character of a nonsolvable group, where the statement of the Theorem 1.1 fails.

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REFERENCES 1. G. Cliff, J. Ritter, and A. Weiss, Group representations and integrality, J. Reine Angew. Math. 426 (1992), 193202. 2. R. Brauer, On the representation of a group of order g in the field of the gth roots of unity, Amer. J. Math. 67 (1945), 461471. 3. E. C. Dade, Characters of groups with normal extra special subgroups, Math. Z. 152 (1976), 131. 4. B. Huppert, ``Endliche Gruppen,'' Vol. I, Springer-Verlag, New YorkBerlin, 1967. 5. I. M. Isaacs, ``Character Theory of Finite Groups,'' Mathematics, Vol. 69, Academic Press, New York, 1976. 6. F. Jonas, Realisierbarkeit von Darstellungen endlicher Gruppen in Einheitswurzelkorpern, Manuscripta Math. 67 (1990), 3539. 7. O. Nei;e, ``Realisierbarkeit von Gruppencharakteren uber Kreiskorpern,'' dissertation, Wi;ner-Verlag, Augsburg, 1995. 8. H. Opolka, Irreducible primitive characters of finite solvable groups, Houston J. Math. 8 (1982), 401407. 9. P. Roquette, Realisierung von Darstellungen endlicher nilpotenter Gruppen, Arch. Math. 9 (1958), 241250. 10. L. Solomon, The representation of finite groups in algebraic number fields, J. Math. Soc. Japan 13 (1960), 144182. 11. H. N. Ward, Representations of symplectic groups, J. Algebra 20 (1972), 182195.

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