Limits of Abelian Subgroups of Finitep-Groups

203, 533]566 Ž1998. JA977340

JOURNAL OF ALGEBRA ARTICLE NO.

Limits of Abelian Subgroups of Finite p-Groups J. L. AlperinU and G. Glauberman† Department of Mathematics, Uni¨ ersity of Chicago, 5734 Uni¨ ersity A¨ enue, Chicago, Illinois 60637 Communicated by Alexander Lubotzky Received July 31, 1997

1. INTRODUCTION Abelian subgroups play a key role in the theory and applications of finite p-groups. Our purpose is to establish some very general results motivated by special results that have been of use. In particular, it is known wKJx that if a finite p-group, for odd p, has an elementary abelian subgroup of order p n, n F 5, then it has a normal elementary abelian subgroup of the same order. Our first main result is a general one of this type. THEOREM A. If A is an elementary abelian subgroup of order p n in a p-group P, then there is a normal elementary abelian subgroup B of the same order contained in the normal closure of A in P, pro¨ ided that p is odd and greater than 4 n y 7. We are very grateful to the referee for an improvement to our argument which gives this linear bound and many other useful comments. The additional information about the location of B is essential for the inductive argument we shall employ. The strategy is to first reduce this theorem to another, namely the following. THEOREM B. If A is an abelian subgroup of order p n in a p-group of exponent p and class at most p q 1, then there is a normal abelian subgroup B of the same order contained in the normal closure of A in P. U †

Supported in part by NSF. E-mail: [email protected]. Supported in part by NSA. E-mail: [email protected]. 533 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

534

ALPERIN AND GLAUBERMAN

The second theorem is now reducible to a question of Lie algebras via Lazard’s functor connecting p-groups and Lie rings; our hypothesis allows use of this functor even though the bound on the class is not p y 1, which is the usual situation. It is now a consequence of the following result. Fix a field F of prime characteristic p. THEOREM C. If A is an abelian subgroup of the Lie algebra L o¨ er F and L is nilpotent of class at most p, then there is an abelian ideal B of L, in the ideal closure of A in L, of the same dimension as A. Our proof of this result is direct but the motivation is entirely from algebraic groups. The analogous result for Lie algebras over the complex numbers is due to Kostant wKos, p. 155x. Let LU be the extension of L to the algebraic closure F U of F. The idea is that in characteristic p one can use the algebraic group corresponding to LU and apply the fixed-point theorem of Borel to the Grassmannian of all subspaces of the Lie algebra LU which are of the same dimension as A. In particular, one could use the form of Borel’s theorem which gives rational fixed points. The authors thank W. Fulton for his helpful comments. We need to introduce some standard notation. For elements or subsets X and Y of a group P we let Ž Y, X; k ., for k non-negative, be defined inductively, as usual, by setting

Ž Y , X ; 0. s Y ;

then Ž Y , X ; 1 . s Ž Y , X .

is the subgroup generated by all commutators of the form Ž y, x . as x runs over X and y runs over Y; and Ž Y, X; i q 1. s ŽŽ Y, X; i ., X .. We now turn to our final main result, which shows that if we are willing to relax the conclusion on normality in Theorem A then we can achieve striking improvements in the bounds on the prime p. THEOREM D. If p is at least 5 and A is an elementary abelian subgroup of order p n of the p-group P, then there is an elementary abelian subgroup B, also of order p n, satisfying the following conditions: Ža. B is normal in its normal closure in P; Žb. Ž P, B; 3. s 1; Žc. B is normalized by any element x of P satisfying

ž

B, x ;

pq1 2

/

s 1.

However, let us occupy the rest of this introductory section with a proof of the reduction of Theorem A to Theorem B.

LIMITS OF ABELIAN SUBGROUPS

535

Proof. The technique is to prove Theorem A by induction and argue until the situation of Theorem B arises. In fact, we shall do a double induction, first on n and then on the order of P. The case n s 1 is trivial so we can proceed with the induction. Again if P has order p n there is nothing to prove. Let Q be a maximal subgroup of P containing A. Thus, there is a normal elementary abelian subgroup A1 of Q, contained in the normal closure of A in Q, also of order p n. The normal closure of A1 in P is contained in the normal closure of A in P so we can assume, by replacing A1 by A, that A is normal in a maximal subgroup of P. In particular, A is normal in its normal closure A P in P. Thus, the series of intersections of A with the upper central series of A P will be a series without any equalities until the central series reaches A P. Thus, A P will be of class at most n, so it will be of exponent p, being generated by elements of order p and having class certainly less than p. By induction on n there is an elementary abelian subgroup B of A P of order p ny 1 normal in P. If B does not coincide with its centralizer in A P, then it is of order p ny 1 contained in the center of a subgroup H of A P, where H is normal in P and has order p n. Since A P is of exponent p, we are done. Hence, we can assume that B is a maximal normal abelian subgroup of A P. Let C s CP Ž B . and N s A P so C l N s B. Since B has order p ny1 we have that B F Zny 1Ž P .. This implies wHup, III.2.11x that B centralizes Pny 1 , the Ž n y 1.st term of the lower central series of P. Consider a commutator w x 1 , . . . , x ny1 x of elements of P with x 1 g N. It lies in Pny 1 l N which is in C l N s B so that we have NrB F Zny2 Ž PrB .. But B F Zny 1Ž P . so N F Z2 ny3 Ž P . and so P2 ny3 centralizes N. Thus, PrCP Ž N . has class at most 2 n y 4. We now form another p-group, the semi-direct product of N by PrCP Ž N . and call it P U . Since N F Z2 ny3 Ž P U . and PrCP Ž N . has class at most 2 n y 4, we have that P U has class at most 4 n y 7. We claim that P U is generated by elements of order p. This is enough to complete the proof. Indeed, then P U will be of exponent p, by our assumption on the size of p wSI, Lemma 4.3.13Žii. and Theorem 4.3.14Žii., pp. 46]47x and Theorem B applies to P U and immediately yields the result. Since N is generated by elements of order p, it suffices to prove that the same holds for PrC p Ž N .. Suppose that x g PrCP Ž N . and y g N. It suffices to prove that x p and y commute. But Ž x p . y s Ž x w x, y x. p and the commutator collection process wHup, III.9x shows this is x p since N is of exponent p and contained in Z2 ny3 Ž P U . while p is large. The rest of the paper is organized in the following way. In Section 2 we study a unipotent automorphism of a non-associative algebra and condi-

536

ALPERIN AND GLAUBERMAN

tions under which the existence of a subalgebra A with A2 s 0 implies the existence of a similar subalgebra of the same dimension which is invariant under the automorphism. In Section 3 we extend this result to a group of automorphisms and derive Theorem C. In the next section we use Lazard’s work to derive Theorems A and B and then in the last section we can apply some of the second author’s earlier work wGG1x to obtain Theorem D.

2. UNIPOTENT AUTOMORPHISMS OF ALGEBRAS In this section we let L be an algebra Žnot necessarily associative. over our field F of prime characteristic p, a a unipotent automorphism of L Žso a y 1 is nilpotent., and we shall be studying the abelian subalgebras A Žthose A for which A2 s 0. of L. We also fix a sequence of a-invariant subspaces 0 s L0 F L1 F . . . , F L s s L such that a induces the identity on each of the successive quotients. Fix a dimension d and define a partial order $ on the set of abelian subalgebras of L of dimension d in the following way. We set B $ BX provided dimŽ B l L i . F dimŽ BX l L i . for all i ) 0, and dimŽ B l L j . - dimŽ BX l L j . for some j. The main result is the following. THEOREM 2.8. If B is maximal under the partial order $ , then B is a-in¨ ariant pro¨ ided the following conditions hold: Ž1. Ž a y 1. p s 0; Ž2. ŽŽ a y 1. i u.ŽŽ a y 1. j ¨ . s 0 for all u and ¨ in B and for all i, j G 1 with i q j G p. It is worth mentioning that the result holds in characteristic zero without the two conditions but we do not need this for any application. One could strengthen the definition of the relation $ by requiring that B l L j - BX l L j instead of dimŽ B l L j . - dimŽ BX l L j . Žsee Proposition 2.2Že... But then $ would no longer be a partial order. To obtain this result, we first obtain a ‘‘limit’’ BU of images of B ŽProposition 2.2.. This is basically a result from algebraic geometry that requires only that L be a vector space, and does not require Condition 2. Then we use Condition 2 to show, in effect, that taking limits induces an isomorphism of B onto BU ŽProposition 2.4Žb. and Theorem 2.6.. Finally, we show that, if B / BU , then B $ BU , a contradiction.

537

LIMITS OF ABELIAN SUBGROUPS

Note that every unipotent linear transformation t of V is invertible because, if Žt y 1. n s 0, 2

1 y Ž t y 1 . q Ž t y 1 . q ??? q Ž y1 . s Ž 1 q Ž t y 1. .

y1

ny1

Ž t y 1.

ny1

s ty1 .

Moreover, if n F p y 1, we can define py1

Log t s

Ý

Ž y1.

iq1

Ž t y 1. i

is1

i

.

Clearly, Log t is nilpotent because ŽLog t . p s 0. Similarly, suppose n is a nilpotent linear transformation of V over F and n p s 0. We define py1

Exp n s

Ý is0

ni i!

.

Then ŽŽExp n . y 1. p s 0, and Exp n is unipotent. The following result appears to be well known, but we have not found a complete published proof. Most of it is proved by finite methods in wGregx. We give a shorter proof that requires infinite methods. PROPOSITION 2.1. Suppose V is a ¨ ector space o¨ er F, t is a unipotent linear transformation of V o¨ er F, and n is a nilpotent linear transformation of V o¨ er F. Assume p

Ž t y 1 . s n p s 0.

Ž 2.1.

Then Ža. ExpŽLog t . and LogŽExp n . are well defined and Exp Ž Log t . s t ,

Log Ž Exp n . s n ,

and Žb.

for e¨ ery natural number r, Exp rn is well defined and r

Exp rn s Ž Exp n . . Proof. Ža. Let Q be the field of rational numbers and A1 be the Q-algebra ww Q X xx of all power series in an indeterminate X. In A1 , define e s X

Xi

Ý iG0

i!

,

e Ž X . s e y 1, X

and

lŽ x. s

Ý Ž y1. iG1

iy1

Xi i!

.

538

ALPERIN AND GLAUBERMAN

Define e Y, eŽ Y ., and l Ž Y . similarly for every Y in A1 such that Y has zero constant term. It is easy to see that these give well defined elements of A1 and that e X and l Ž X . are the familiar Maclaurin series for the functions e t and logŽ t y 1.. It is shown in wB, p. A.IV.40x that l Ž eŽ X .. s eŽ l Ž X .. s X. We take the natural homomorphism C of A1 onto the algebra A2 s Qw X xrŽ X p . , i.e., the quotient of the polynomial algebra Qw X x by the principal ideal generated by X p. For X s C Ž X . and for functions e and l on A 2 defined like eŽ X . and l Ž X . above, l Ž eŽ X . . s eŽ l Ž X . . s X . Let Zw1rŽ p y 1.!x be the subring of Q obtained by adjoining the number 1rŽ p y 1.! to the ring of integers Z, and let A 3 be the subring of A 2 generated by Z 1r Ž p y 1 . !

and

X.

Then the functions e and l on A 2 map A 3 into A 3 . Now we take the natural homomorphism CX of A 3 into the algebra over F spanned by n , with X mapping to n . We define the functions e and l similarly on the image of XA 3 under CX . Then, using our previous definitions, we have

n s l Ž e Ž n . . s l Ž exp n y 1 . s log Ž exp n . . A similar homomorphism of A 3 with X mapping to t y 1 yields

t y 1 s e Ž l Ž t y 1 . . s e Ž log t . s y1 q exp Ž log t . , and thus t s expŽlog t .. Žb. The proof is similar to that of Ža.. For commuting indeterminate X, Y and the algebra of formal power series Qw X, Y x, it is proved in wB, pp. A.IV.39]40x that e Xq Y s e X e Y . By the method of Ža., this yields a proof of Žb. by induction on r. Now take an indeterminate t over F, and let F Ž t . be the field obtained by adjoining t to F. For a vector space V over F, let V Ž t . be the vector space F Ž t . mF V over F Ž t .. PROPOSITION 2.2. Suppose l is a linear transformation of a ¨ ector space V o¨ er F, and l p s 0. For each u g V, let py1

t t Ž u. s

Ý is1

li Ž u . i!

ti

Ž in V Ž t . . .

Ž 2.2.

539

LIMITS OF ABELIAN SUBGROUPS

Suppose U is a subspace of V. Let s s dim U, and take a basis u1 , . . . , u s of U. On the exterior product space Ls Ž V Ž t .., let s Ž py1 .

c Ž t . s t t Ž u1 . n t t Ž u 2 . n ??? n t t Ž u s . s

Ý

ck t k ,

Ž 2.3.

ks0

where c k g Ls Ž V . for each k. Then c 0 s u1 n u 2 n ??? n u s . Take the largest integer e for which c e / 0. Then the following conditions are satisfied: Ža. Up to scalar multiples, cŽ t . is independent of the choice of the basis u1 , . . . , u s . Žb. For some linearly independent ¨ ectors uU1 , uU2 , . . . , uUs in V, uU1 n uU2 n ??? n uUs s c e . Žc. The subspace U U s ² uU1 , uU2 , . . . , uUs : of V has dimension s and is in¨ ariant under l. Žd. For each l-in¨ ariant subspace X of V, dim U U l X G dim U l X . Že. Suppose 0 s V0 F V1 F ??? Vm s V is an increasing series of subspaces of V such that

lVi F Viy1 ,

for i s 1, 2, . . . , m.

Ž 2.4.

If U is not in¨ ariant under l, then U U l Vi ) U l Vi

for some i.

Remark. By Žc. and Žd., U U is contained in the smallest l-invariant subspace of V containing U, i.e., in U q l Ž U . q l2 Ž U . q ??? ql py 1 Ž U . . Proof. Ža. Since

t t Ž u q ¨ . s t t Ž u. q t t Ž ¨ . ,

and

t t Ž au . s at t Ž u .

for every u, ¨ g V and a g F, this follows from the standard properties of the exterior product.

540

ALPERIN AND GLAUBERMAN

Žb. Let h1Ž t ., . . . , h s Ž t . be polynomials in t with coefficients from V chosen such that h1 Ž t . n h 2 Ž t . n ??? n h s Ž t . s c Ž t .

Ž 2.5.

and Ý deg h i is minimal subject to Ž2.5.. For each i, let d i be the degree of h i? and uUi be the coefficient of t d i in h i Ž t .. Set d s Ý d i . Clearly, from Ž2.5., uU1 n uU2 n ??? n uUs is the coefficient of t d in c Ž t .

Ž 2.6.

and is equal to c e if uU1 n uU2 n ??? n uUs / 0 Žin which case d s e .. Suppose uU1 , . . . , uUs are linearly dependent. Let a1 uU1 q ??? qa s uUs s 0, where a i g F for each i and a i / 0 for some i. Among all subscripts i for which a i / 0, choose one, i 0 for which d i has the highest degree. We may assume that i 0 s 1 and that a1 s 1. Let hU1 Ž t . s h1 Ž t . q

a i t d1yd i h i Ž t . .

Ý 2FiFs

Then hU1 n h 2 Ž t . n h 3 Ž t . n ??? n h s Ž t . s cŽ t ., but deg hU1 q

deg h i - deg h1 q

Ý 2FiFs

Ý

deg h i ,

2FiFs

contrary to the choice of h1Ž t ., . . . , h s Ž t .. This contradiction shows that uU1 , . . . , uUs are linearly independent. Now Žb. follows from Ž2.6.. Žc. From Žb., U U has dimension s. Since l p s 0, we may let py1

t s t 1 s Exp l s

Ý is1

li i!

.

Then t operates on V and, in an obvious way, on V Ž t .. From Ž2.2., we may describe t t Ž u. as ŽExp l t .Ž u. for each u g V. The usual calculations for multiplying exponentials show that, for u g V, py1

t Žt t Ž u . . s

Ý is1

li i!

py1

ž

Ý js0

l j Ž u. j!

py1

/

tj s

Ý ks0

lk Ž u . k!

k Ž t q 1 . s t tq1 Ž u . .

541

LIMITS OF ABELIAN SUBGROUPS

Therefore, from Ž2.3. and Ž2.6., e

Ý t Ž c k . t k s t Ž c Ž t . . s t Žt t Ž u1 . n t t Ž u 2 . n ??? n t t Ž u s . . ks0

s t tq1 Ž u1 . n t tq1 Ž u 2 . n ??? n t tq1 Ž u s . e

s c Ž t q 1. s

Ý ck Ž t q 1. k , ks0

and Žby comparison of coefficients of t e .

t Ž uU1 n uU2 n ??? n uUs . s t Ž c e . s c e s uU1 n uU2 n ??? n uUs . Hence, t ŽU U . s U U . Since py1

l s Log Ž Exp l . s Log t s

Ý Ž y1.

iq1

is1

Ž t y 1.

i

i

,

we have lŽU U . F U U . Žd. Let r s dimŽU l X .. We may assume that u1 , . . . , u r g X. Let us apply the process in the proof of Žb. to X and U l X in place of V and U. We obtain polynomials f 1Ž t ., . . . , f r Ž t . in t with coefficients from X such that f 1 Ž t . n f 2 Ž t . n ??? n f r Ž t . s t t Ž u1 . n t t Ž u 2 . n ??? t t Ž u r .

Ž 2.7.

and the leading coefficients uU1 , . . . , uUr of f 1Ž t ., . . . , f r Ž t . are linearly independent. From Ž2.3. and Ž2.7., and the associativity of the exterior algebra of V Ž t ., f 1 Ž t . n ??? f r Ž t . n t t Ž u rq1 . n ??? t t Ž u s . s c Ž t . . Now we choose polynomials h1Ž t ., . . . , h s Ž t . in t with coefficients from V such that Ý deg h i is minimal subject to h1 Ž t . n ??? n h s Ž t . s c Ž t . , and the leading coefficients of h1 Ž t . , . . . , h s Ž t . span a subspace of V that contains uU1 , . . . , uUr .

Then h1 , . . . , h s exist, since f 1 Ž t . , f 2 Ž t . , . . . , f r Ž t . , t t Ž u rq1 . , . . . , t t Ž u s .

Ž 2.8.

542

ALPERIN AND GLAUBERMAN

satisfy Ž2.8.. The proof of Žb. can be adopted to this case; note that, in that proof, the leading coefficient of h1 is a linear combination of the leading coefficients of h 2 , . . . , h s , so that replacing h1 by hU1 either preserves or enlarges the subspace of V spanned by the leading coefficients. Thus, ² uU1 , . . . , uUr : F U U l X , and dim U U l X G r s dim U l X. Že. Assume U is not invariant under l. Take i to be maximal subject to U U l Vi s U l Vi .

Ž 2.9.

U U l Viq1 / U l Viq1 .

Ž 2.10.

Then 0 F i - m and

Suppose U l Viq1 F U U l Viq1. By Ž2.10., U U l Viq1 ) U l Viq1 , as required. Next, suppose U l Viq1 is not contained in U U l Viq1. Take u g U l Viq1 such that u f U U l Viq1. Then u f U U and l p Ž u. s 0. Take r minimal so that l r Ž u. s 0. Let X s² u, l Ž u . , l2 Ž u . , . . . , l ry1 Ž u .: . Then X is invariant under l. By Žd., dim U U l X G dim U l X . Since u g U l X and u f U U , there exists ¨ g U U l X such that ¨ f U. By Ž2.4.,

lVj F Vjy1

for j s 1, 2, . . . , m.

Ž 2.11.

Therefore, each subspace Vj is invariant under l, and lŽ u. g lŽ Viq1 . F Vi . Hence,

lŽ u . , l2 Ž u . , . . . , l ry1 Ž u . g Vi . Since ¨ g U U and ¨ f U, Ž2.9. shows that ¨ f Vi . So ¨ s a0 u q a1 l Ž u . q ??? qa ry1 l ry1 Ž u . ,

with a0 , a1 , . . . , a ry1 g F and a0 / 0. Replacing ¨ by ay1 0 ¨ , we may assume that a0 s 1.

LIMITS OF ABELIAN SUBGROUPS

543

As U U is invariant under l, we get

l ry1 Ž u . s l ry1 Ž ¨ . g U U . Similarly,

l ry2 Ž u . s l ry2 Ž ¨ . y a1 l ry1 Ž u . g U U . By an obvious induction,

l ry3 Ž u . , l ry4 Ž u . , . . . , lŽ u . ,

u g UU ,

contrary to u f U U . This contradiction completes the proof of Že. and of Proposition 2.2. We now consider the following situation: HYPOTHESIS 2.3. Ža.

L is a non-associati¨ e algebra o¨ er a field F of prime characteristic

p; Žb. a is an automorphism of L; Žc. Ž a y 1. p s 0; Žd. l s Log a ; Že. U is a subspace of L; Žf. moreo¨ er,

Ž Ž a y 1. i u .Ž Ž a y 1. j ¨ . s 0, for all u, ¨ g U and all natural numbers i, j such that i q j G p. Note that, by Žc., l is well defined and l p s 0. For applications, usually L will be a Lie algebra. PROPOSITION 2.4. Assume Hypothesis 2.3. Let t be an indeterminate o¨ er F, and let F Ž t . be the field obtained by adjoining t to F. Let L Ž t . s F Ž t . mF L, and regard LŽ t . as an algebra o¨ er F Ž t . in the usual way. Define t t on LŽ t . as in Proposition 2.2, i.e., py1

tt s

Ý is0

li Ž u . i!

t i,

for each u g L.

544

ALPERIN AND GLAUBERMAN

Then Ža.

for each natural number k, k

l k Ž u¨ . s

Ý is0

k i

ž /l Ž i

u . l kyi Ž ¨ . ,

for all u, ¨ g U,

and

t t Ž u¨ . s t t Ž u . t t Ž ¨ .

Žb.

for all u, ¨ g U.

Proof. By Proposition 2.1, a r s Exp r l for every natural number r, including r s 1. Note that we may think of t t as Exp t l. For u, ¨ g U and each natural number r, py1

r kl k Ž u¨ .

Ý ks0

k!

s Ž Exp r l . Ž u¨ . s a r Ž u¨ . s a r Ž u . ? a r Ž ¨ . r ili Ž u .

py1

s

žÝ

/ž Ý

i!

is0

r jl j Ž ¨ .

Ý Ý

i! j!

is0 js0

/

j!

js0

r iqjli Ž u . l j Ž ¨ .

py1 py1

s

py1

.

Ž 2.12.

Ža. Take u, ¨ g U. By Hypothesis 2.3, l p s 0 and i j Ž a y 1. u ? Ž a y 1. ¨ s 0

whenever i, j G 1 and i q j G p. Since

l s Log a s

Ž y1.

Ý

iq1

Ž a y 1.

i

i

1FiFpy1

,

it follows that

li Ž u . ? l j Ž ¨ . s 0

whenever i , j G 1 and i q j G p.

Ž 2.13.

Now we may rewrite Ž2.12. as py1

r kl k Ž u¨ .

Ý

k!

ks0

py1

s

Ý

rk

ks0

k

li Ž u . l kyi Ž ¨ .

Ý

i! Ž k y i . !

is0

,

and then as py1

Ý ks0

rk

½

l k Ž u¨ . k!

y

k

li Ž u . l kyi Ž ¨ .

Ý

i! Ž k y i . !

is0

5

s 0.

Ž 2.14.

545

LIMITS OF ABELIAN SUBGROUPS

We may think of r as a natural number or as an element of F; note that 0 0 s p 0 s 1. Let M s Ž m sr . be the Vandermonde matrix of degree p given by m sr s r s ,

for s, r s 0, 1, . . . , p y 1 Ž where 0 0 s 1 . .

Let y be the row vector Ž y 0 , . . . , y py1 . given by yk s

l k Ž u¨ . k!

y

k

li Ž u . l kyi Ž ¨ .

Ý

i! Ž k y i . !

is0

.

Since Ž2.14. holds for all r, the product yM is the zero row vector of length p. As a matrix M is nonsingular, y is the zero row vector. Therefore, for k s 0, 1, . . . , p y 1, k

0 s k! y k s l k Ž u¨ . y

Ý is0

k i

ž /l Ž i

u . l kyi Ž ¨ . .

This proves Ža. for k F p y 1. Since l p s 0, Ž2.13. yields Ža. for k G p. Žb. Let u, ¨ g U. Here, in effect, we reverse the proof of Ža.. We have py1

tt Ž u. tt Ž ¨ . s

žÝ

t ili Ž u . i!

is0

py1 py1

s

py1

/ž Ý

t jl j Ž ¨ . j!

js0

t iqjli Ž u . l j Ž ¨ .

Ý Ý

i! j!

is0 js0

/

.

By Ž2.13., we can write py1

tt Ž u. tt Ž ¨ . s

Ý ks0

k

t

k

Ý is0

li Ž u . l kyi Ž ¨ . i! Ž k y i . !

py1

s

Ý ks0

tk k!

k

Ý is0

k i

ž /l Ž i

u . l kyi Ž ¨ . .

By Ža., py1

t t Ž u. t t Ž ¨ . s

Ý ks0

tk k!

l k Ž u¨ . s t t Ž u¨ . .

This proves Žb.. For later use, we now state what is roughly a converse to Proposition 2.4. Recall that a deri¨ ation of L over F is an endomorphism d of L over F such that

d Ž xy . s d Ž x . y q x d Ž y . ,

for all x, y g L.

546

ALPERIN AND GLAUBERMAN

PROPOSITION 2.5. Ža.

Assume Hypothesis 2.3. Suppose d is a deri¨ ation of L.

For e¨ ery natural number k, k

d k Ž xy . s

Ý is0

Žb.

k i

ž /d Ž i

x . d kyi Ž y . ,

for all x, y g L.

Assume d p s 0 and

d i Ž u . d j Ž ¨ . s 0, for all u, ¨ g U and all i, j G 1 such that i q j G p. Then Exp d exists and

Ž Exp d . Ž u¨ . s Ž Exp d . Ž u . ? Ž Exp d . Ž ¨ . , for all u, ¨ g U. Proof. Ža. This follows from induction on k and the use of the identity

ž

kq1 k k s q , i i iy1

/ ž/ ž

/

1 F i F k.

Žb. As discussed before Proposition 2.1, Exp d is well defined, unipotent, and hence invertible, and p

Ž Ž Exp d . y 1 . s 0

and

d s Log Ž Exp d . .

Let b s Exp d . By Ža., d satisfies the conditions on l in Proposition 2.4Ža.. The proof of Proposition 2.4Žb. shows that

b Ž u¨ . s b Ž u . b Ž ¨ . ,

for all u, ¨ g U.

ŽHowever, b need not be an automorphism of L if U / L.. THEOREM 2.6.

Assume Hypothesis 2.3. Suppose U 2 s 0, i.e., u¨ s 0,

for all u, ¨ g U.

Define a subspace U U of L by letting V s L in Proposition 2.2. Then U U 2 s 0 and U U is in¨ ariant under a . Remark. Of course, U U also satisfies conclusions Ža. ] Že. of Proposition 2.2.

547

LIMITS OF ABELIAN SUBGROUPS

Proof. Let us recall our notation from Proposition 2.2. We let s s dimŽU . and choose a basis u1 , u 2 , . . . , u s of U. Take t, F Ž t ., LŽ t ., and t t as in Proposition 2.4. In the exterior product space Ls Ž LŽ t .., let s Ž py1 .

c Ž t . s t t Ž u1 . n t t Ž u 2 . n ??? n t t Ž u s . s

Ý

ck t k ,

ks0

where c k g L L. for each k. Let e be the largest subscript for which c e / 0. Let sŽ

t t Ž U . s t t Ž u . u g U 4 . Since t t is a vector space endomorphism of LŽ t . over F Ž t .,

t t Ž U . s² t t Ž u1 . , . . . , t t Ž u s .: . As U 2 s 0, Proposition 2.4Žb. yields

Žt t Ž U . .

2

s 0.

Ž 2.15.

In the proof of Proposition 2.2, we found polynomials h1Ž t ., . . . , h s Ž t . in LŽ t ., and a basis uU1 , . . . , uUs of U U such that h1 Ž t . n ??? n h s Ž t . s c Ž t . s t t Ž u1 . n ??? n t t Ž u s . and, for i s 1, . . . , s with d i s deg h i , uUi is the leading coefficient Ž the coefficient of t d i . of h i Ž t . . Then h1Ž t ., . . . , h s Ž t . form a basis of t t ŽU . over F Ž t . and, for each i, j, uUi uUj s coefficient of t d iqd j in h i Ž t . h j Ž t . s 0. This shows that U U 2 s 0. Since U U is invariant under l and a s Exp l, U U is invariant under a .

We give counterexamples to some generalizations of Theorem 2.6 in Example 3.6. Remark 2.7. An examination of the proof of Theorem 2.6 shows that, by strengthening Hypothesis 2.3Žf., we may replace the condition U2 s 0 by U 2 ? U s 0, or U 2 ? U s U ? U 2 s 0, or various identities, and obtain the analogous condition for U U .

548

ALPERIN AND GLAUBERMAN

Now we come to the proof of the main result of this section Žwhich is stated at the beginning of the section.. Proof. Recall that Ž a y 1. p s 0. Therefore, we may define

l s Log a . Observe that Hypothesis 2.3 is satisfied for U s B. Now define a subspace U U of L by letting V s L in Proposition 2.2. By Theorem 2.6, UU 2 s 0

U U is invariant under a .

and

By parts Žc., Žd., and Že. of Proposition 2.2, U $ U U if U is not invariant under a . However, we chose B to be maximal under $ . Therefore, B is invariant under a , as desired.

3. UNIPOTENT GROUPS OF AUTOMORPHISMS OF ALGEBRAS Throughout this section, F is a field of prime characteristic p. ŽAs mentioned above, our results can be extended to characteristic zero.. In this section, we extend our main result on a single unipotent automorphism ŽTheorem 2.8. to a result about a group of automorphisms ŽTheorem 3.1.. We also apply the results of Section 2 to Lie algebras in Theorem 3.1 and Corollary 3.4 Žwhich gives Theorem C.. The rest of this section consists of counterexamples ŽExamples 3.5 and 3.6. to generalizations of Theorems 3.1 and 3.3. The reader interested only in groups may skip this section. Our next result is an immediate consequence of Theorem 2.8. THEOREM 3.1. Suppose G is a group of automorphisms of a finite-dimensional algebra L o¨ er F, and 0 s L0 F L1 F ??? F L s s L is a sequence of subspaces of L such that

Ž a y 1 . L i F L iy1 ,

for i s 1, 2, . . . , s and e¨ ery a g G.

Ž 3.1.

Let B0 be a subspace of L for which B02 s 0. Define a partial ordering $ on the set of all subspaces B of L for which B2 s 0

and

dim B s dim B0

LIMITS OF ABELIAN SUBGROUPS

549

by setting B $ BX if Ži. dim B l L i F dim BX l L i for i s 1, 2, . . . , s, and Žii. dimŽ B l L i . - dimŽ BX l L i ., for some i Ž1 F i F s .. Take BU maximal under $ . Let n be a natural number. Assume that G is generated by a set S of elements a such that p

Ž a y 1. s 0

Ž 3.2.

Ž Ž a y 1. i u .Ž Ž a y 1. j ¨ . s 0

Ž 3.3.

and

for all u, ¨ g BU and all i, j G 1 such that i q j G p. Then BU is in¨ ariant under G. Remark 3.2. As in Remark 2.7, one may generalize this result by replacing the condition B 2 s 0 by B 2 ? B s 0 or various other identities Žand strengthening condition Ž3.3... Recall that an abelian subalgebra of a Lie algebra L is a subalgebra B such that w B, B x s 0. Notation. For elements x, y in a Lie algebra, define the iterated commutator w y, x; i x for each natural number i inductively by

w y, x ; 1 x s w y, x x and

w y, x ; i q 1 x s w y, x ; i x , x ,

for each i s 1, 2, 3, . . . .

For subalgebras X, Y, define w Y, X; i x to be the subspace spanned by all elements w y, x; i x for x g X, y g Y. THEOREM 3.3. Suppose L is a finite-dimensional nilpotent Lie algebra o¨ er F and B0 is an abelian subalgebra o¨ er L. Let 0 s L0 F L1 F ??? F L s s L

Ž 3.4.

be a central series for L, and let Bˆ0 s B0 [

Ý w B0 , L; i x . iG1

Define a partial ordering $ as in Theorem 2.8 Ž or Theorem 3.1. on the set of all abelian subalgebras B of L for which B F Bˆ0 U

and

and take B maximal under $ .

dim B s dim B0 ,

550

ALPERIN AND GLAUBERMAN

Assume that L is generated by a set E of elements x such that

w y, x ; p x s 0 s w u, x ; i x , w ¨ , x ; j x

Ž 3.5.

for all y g L and u, ¨ g BU , and all natural numbers i, j such that i q j G p. Then BU is an ideal of L. Proof. Take E as given. Because of the Jacobi identity, it suffices to show that w BU , x x ; BU for every x g E. So take an arbitrary element x of E. Let K be the subspace of L spanned by BU and all iterated commutators

w u, x ; i x ,

u g BU ,

i G 1.

It is easy to see, by using this spanning set, that If p ) 0, then Ž 3.5. is valid with BU replaced with K .

Ž 3.6.

Note that K F Bˆ0 . Now let M be the vector space external direct sum K [ w KK x, i.e., M s  Ž y, z . y g K , z g w KK x 4 . Then M is a vector space over F in the usual way. Define multiplication on M by

Ž y, z . , Ž yX , zX . s Ž 0, w y, yX x . . It is easy to see that M becomes a Lie algebra over F under this definition. Moreover, for every abelian subalgebra B of L, the subspace B [ w KK x s  Ž y, z . y g B, z g w KK x 4 is an abelian subalgebra of M.

Ž 3.7.

Recall that the adjoint of x is the linear mapping on L given by

Ž ad x . Ž y . s w y, x x , and that ad x is a derivation of L. Define d : M ª M by

d Ž y, z . s Ž Ž ad x . Ž y . , Ž ad x . Ž z . . s Ž w y, x x , w z, x x . . It is easy to see that d is a derivation of M. Since Ž3.4. is a central series of L,

Ž ad x . Ž L i . s w L i , x x F L iy1 ,

i s 1, 2, . . . , s.

Ž 3.8.

LIMITS OF ABELIAN SUBGROUPS

551

From Ž3.6., it is easy to see that

dps0

and

w wd i , wXd j x s 0,

Ž 3.9.

for all w, wX in M and all natural numbers i, j such that i q j G p. Let a and l be the identity and zero mappings of M and let U s M. Then Hypothesis 2.3 is satisfied with M in place of L. By Ž3.9. and Proposition 2.5Žb., Exp d is defined and is an automorphism of M. Now we change notation to let a be Exp d and l be d , and let U be BU [ w KK x. Let V0 s 0

Vi s L iy1 [ w KK x , i s 1, 2, . . . , s q 1.

and

Note that lŽ Vi . s d Ž Vi . F Viy1 for each i. Take U U as in Proposition 2.2 Žwith V s M .. From Proposition 2.2Žd., Že. and our maximal choice of BU , we have U U s U s BU [ w KK x . Therefore, by Theorem 2.6, U U is invariant under a , and hence under d . It follows that BU G Ž ad x . Ž BU . s w BU , x x as desired. COROLLARY 3.4. Suppose L is a finite-dimensional nilpotent Lie algebra o¨ er F and B is an abelian subalgebra of L. Assume one of the following conditions: Ži .

L is generated by elements x such that

w y, x ; p x s 0 s w u, x ; i x , w ¨ , x ; p y i x , for all y, u, ¨ g L and i s 1, 2, . . . , p y 1, Žii. L is generated by elements x such that

w y, x ; n x s 0

for n s

pq1 2

and all y g L;

or Žiii.

L has nilpotence class at most p.

Then there exists an abelian ideal BU of L for which BU F B q

Ý w B, L; i x iG1

and

dim BU s dim B.

552

ALPERIN AND GLAUBERMAN

Proof. Note that L satisfies Ži., since Žii. and Žiii. are clearly stronger than Ži.. However, Ž3.5. follows from Ži.. Take BU as in Theorem 3.3 Žwith B in place of B0 .. Proof of Theorem C. Clearly Theorem C follows from Corollary 3.4. EXAMPLE 3.5. We give an example to show that Theorem 3.3, Corollary 3.4, and Theorem C are false, if one deletes the hypotheses on L. Let F s Fp . Let r be an integer, given by r s 3 if p s 2;

r s 6 if p s 3;

rs

pq3 2

if p G 5.

Let M be the Lie algebra over F given by the generators u 0 , u1 , . . . , u r subject to the relations

w x, y x , z s 0

for all x, y, z g M.

Ž 3.10.

Then w M, M x has a basis given by

w ui , u j x ,

0 F i - j F r.

Ž 3.11.

Define u i s 0 for i s r q 1, r q 2, . . . . We may define a linear transformation D on M by Du i s u iq1

D w u i , u j x s w u iq1 , u j x q w u i , u jq1 x Ž 0 F i - j F r . .

and

Thus Du r s 0 and Dw u i , u r x s w u iq1 , u r x for i s 0, 1, . . . , r y 1. Clearly, D is a derivation of M. By Proposition 2.5, k

D k w u0 , u2 x s

Ý is0

k i

ž/

k

D i u 0 , D kyi u 2 s

Ý is0

k i

ž /w

u i , u kq2yi x , Ž 3.12.

for each positive integer k. If k G 2 r y 2, then each summand has iGrq1

or

kq2yiGrq1

or

i s k q 2 y i s r,

whence w u i , u kq1yi x s 0; thus D u 0 , u 2 x s 0. Let M1 be the subspace of M spanned by w u 0 , u 2 x and D k w u 0 , u 2 x for k s 1, 2, . . . 2 r y 3. Since M1 is contained in the center of M, M1 is an ideal of M. Let L0 be the quotient MrM1. Then D induces a derivation D 0 on L0 . For each i s 0, 1, 2, . . . , let x i be the element u i q Mi in L0 . kw

LIMITS OF ABELIAN SUBGROUPS

553

We claim that w x ry1 , x r x / 0. To see this, we need only show that w u ry1 , u r x is not in the subspace M1. We use the basis Ž3.11. of w M, M x. By Ž3.12., the only integer for which w u ry1 , u r x can possibly appear with nonzero coefficient in D k w u 0 , u 2 x is 2 r y 3 Žsince i q Ž k q 2 y i . s k q 2 for all i .. Since u j s 0 whenever j G r q 1, we need only consider summands for which i F r and k q 2 y i F r ,

i.e., r y 1 s k q 2 y r F i F r .

So D 2 ry3 w u 0 , u 2 x s

ž

2 ry3 3 w u , u x ss w u , u x , w u ry1 , u r x q 2 r y r ry1 ry1 r r ry1

/

ž

/

for ss

ž

2r y 3 y 2r y 3 . r ry1

/ ž

/

If p s 2, we have rs3

ss

and

3 3 y s 3 y 1 s 2 s 0 in F 2 . 2 3

ž/ ž/

If p s 3, then rs6

ss

and

9 9 y s 3 2 ? 14 s 3 ? 28 s 0 in F 3 . 5 6

ž/ ž/

If p s 5 then rs

pq3 2

,

1 F r y 1 - r F p y 1, 2 r y 3 s p,

and hence ss

p p y s 0. r ry1

ž / ž/

ŽMore generally, D p s 0 whenever p G 5.. Thus, in all cases,

w x ry1 , x r x / 0. Now let L be the semi-direct product L0 [ FD 0 , and let L1 s w L0 , L0 x, d s dim L1 , and B s Fx 0 [ Fx 2 [ L1. Then B is an abelian subalgebra of L and dim B s 2 q d.

554

ALPERIN AND GLAUBERMAN

Suppose L has an abelian ideal BU of dimension 2 q d. Since L1 is clearly an ideal of L, we obtain an abelian ideal

Ž BU q L1 . rL1 F LrL1 . Because D 0 u i s u iq1 for i s 0, 1, . . . , r y 1, it is easy to see that the only nonzero D 0-invariant subspaces of L0rL1 are

Ž ² x i , x iq1 , . . . , x r : [ L1 . rL1 ,

i s 1, 2, . . . , r .

Therefore, ŽŽ BU l L0 . [ L1 .rL1 is one of these, or is 0. Since L1 is contained in the center of L0 and w x ry1 , x r x / 0, BU l L0 F Ž BU l L0 . [ L1 s L1

² x r : [ L1 .

or

Now dim² x r : [ L1 s 1 q d s dim BU y 1, and LrL0 is 1-dimensional. Therefore, BU l L0 s ² x r : [ L1

BU g L0 .

and

So there exists y g BU y Ž BU l L0 .. But then L1 lies in the center of ² L0 , y : s L. This is impossible, since

w x ry2 , x r x g L1

and

D 0 w x ry2 , x r x s w x ry1 , x r x q w x ry2 , x rq1 x s w x ry1 , x r x / 0. This contradiction shows that L has no abelian ideal of dimension d q 2, although it has an abelian subalgebra of dimension d q 2. Therefore, Theorem 3.3, Corollary 3.4, and Theorem C do not hold in general without some restriction like Ž3.5.. We give counterexamples to a generalization of some of our other results, based on an example by the first author wHup, Aufg. 31, p. 349x. EXAMPLE 3.6. Suppose p is a prime and F s Fp . Let L be the Lie algebra over F with generators x1 , x 2 , . . . , x p ,

y1 , y 2 , . . . , yp

and relations

w x i , yi x s 0 Ž i s 1, . . . , p . ,

w x, y x , z s 0 for all x, y, z g L.

LIMITS OF ABELIAN SUBGROUPS

555

Let LX s w L, L x. One sees that LX has basis w x i , x j x , w yi , y j x , w x i , yiX x

Ž 1 F i , iX , j X F p . ,

subject to i - j and i / iX ; p s 2 p 2 y 2 p, dim L s 2 p 2 ; 2 L has a unique automorphism a such that a x i s x iq1 , yia s yiq1 Ž for i mod p . , and a p s 1; dim LX s 4

ž/

and the only abelian subalgebras of L of dimension 2 p 2 y 2 p q 2 are ² x i , yi , LX : Ž i s 1, 2, . . . , p ., and none is invariant under the group ² a :. Let G s ² a :. In this example, Ž a y 1. p s a p y 1 s 0, but L and G violate the conclusion Žand hence Ž3.3.. in Theorem 3.1. The proof of Theorem 3.1 and its predecessors show that this is also a counterexample to some generalization of Proposition 2.4 and Theorems C and 2.6.

4. APPLICATIONS TO p-GROUPS In this section, we transform Theorem 2.8, a result about algebras, into Theorem 4.5, a result about p-groups. Then we prove a special case of Theorem B that allows us to prove Theorem A. Our main tool for applying the results of the previous section to p-groups is the Baker]Campbell]Hausdorff Formula wJ, pp. 170]174x. As usually stated, one starts with a Lie algebra L over a field F of characteristic 0, and turns L into a group by defining, for x, y g L, xy s x q y q

1 2

w x, y x q ??? .

Ž 4.1.

Some restrictions are necessary to make the series converge, e.g., L may be nilpotent. However, M. Lazard showed in wL, Theorem 4.3x that Ž1. Equation Ž4.1. can be used to define a group if F has some prime characteristic p and L is nilpotent of class at most p y 1; and Ž2. in many situations, one may start with a group G and invert Ž4.1. to convert G into a Lie algebra. Our approach here is to apply Ž2. and then use Theorem 2.8 to assert that a particular abelian subgroup of G is normal in G, or invariant under some group of automorphisms of G. Such applications for groups, defined over infinite fields appear to be well known, and then one may apply other methods; e.g., Kostant used the Borel Fixed Point Theorem for his result

556

ALPERIN AND GLAUBERMAN

mentioned in the Introduction. For this reason, we restrict our attention to finite p-groups. Henceforth in this section, assume that P is an arbitrary finite p-group for an arbitrary prime p, and all groups are finite. Consider the following conditions on a triple Ž Q, R, A.. HYPOTHESIS 4.1. Ž4.1a. of Q. Ž4.1b.

R is a p-group, Q 1 R, and A is an elementary abelian subgroup There exists a central series of Q, 1 s Q Ž 0 . F Q Ž 1 . F ??? F Q Ž s . s Q,

which can be extended to a central series of R, such that whene¨ er B is an elementary abelian subgroup of Q and < B < s < A<

B l Q Ž i . G A l Q Ž i . for all i Ž 0 F i F s .

and

then there is no ¨ alue of i Ž0 F i F s . for which B l QŽ i . ) A l QŽ i .. PROPOSITION 4.2. Hypothesis 4.1.

Suppose a triple Ž Q, R, A 0 . satisfies condition Ž4.1a. in

Ža. For e¨ ery central series Q that can be extended to a central series of R, there exists an elementary abelian subgroup A of Q such that < A < s < A 0 < and Ž Q, R, A. satisfies Hypothesis 4.1, and Žb. for some choice of the central series Q and of A in Ž a., A F ² A 0R :. Proof. Ža. Consider the set of all elementary abelian subgroups of Q of order < A 0 <. Put a partial ordering on this set similar to the partial ordering in Theorem 2.8 Žwith order in place of dimension.. Then take a maximal element A such that A 0 $ A or A 0 s A. Žb. Extend the series 1 F ² A 0R : F Q F R to a central series of R, and take the portion from 1 to Q as the central series in Hypothesis 4.1Žb.. Then take A as in Ža.. We obtain < A l ² A 0R :< G < A 0 l ² A 0R :< s < A 0 < s < A < . Therefore, A F ² A 0R :. Now we state the special case of Lazard’s results that we need: THEOREM 4.3 wL, Theoreme ´ ` II.4.6, pp. 179]180x. Suppose P has exponent 1 or p and nilpotence class at most p y 1. Then, by means of an in¨ ersion of the formula Ž4.1., P may be regarded as a Lie algebra o¨ er Fp . For

557

LIMITS OF ABELIAN SUBGROUPS

these structures, the notions of subgroup Ž respecti¨ ely, normal subgroup. and of subalgebra Ž respecti¨ ely, ideal . coincide. Moreo¨ er, for x, y, g P and any integer n, x n in the group is equal to nx in the algebra, and xy s yx in the group if and only if w x, y x s 0 in the algebra. Note. The second paragraph of Theorem 4.3 is not stated explicitly in the result quoted, but is easy to derive from Ž4.1., e.g., as discussed in wL, pp. 142]143x. Note that each automorphism of P as a group becomes an automorphism of P as a Lie algebra and, in particular, becomes a linear transformation of P. Notation. We recall some notation from Section 1. For x, y g P, let

Ž x, y . s xy1 yy1 xy. ŽThis disagrees with wL, p. 105x, but we will not make further references to wLx.. Then, we define Ž y, x; i . for each positive integer i inductively by

Ž y, x ; 1 . s Ž y, x .

and

Ž y, x ; i q 1 . s Ž Ž y, x ; i . , x . for each i s 1, 2, 3, . . . . ŽThis is analogous to our definition of iterated commutators w ¨ , u; i x for u, ¨ in a Lie algebra, before Theorem 3.3.. For x g P and a subgroup Q of P, let Ž Q, x . be the subgroup of P generated by all the elements Ž y, x . for y g Q. LEMMA 4.4. Suppose Q F P and g g P. Assume that Q has exponent 1 or p and nilpotence class at most p y 1, and g normalizes Q. Let a be the automorphism on Q gi¨ en by conjugation by g in the group P, i.e.,

a Ž x . s gy1 xg. Consider a as a linear transformation of Q, for Q represented as a Lie algebra. Then Ža. Ž Q, g . is normalized by Q, and Žb. Ž a y 1. Q is contained in Ž Q, g .. Proof. Ža. For x, y g Q, y

Ž xy, g . s yy1 xy1 x g y g s yy1 xy1 x g yyy1 y g s Ž x, g . Ž y, g . , so y1 Ž x, g . s Ž xy, g . Ž y, g . g Ž Q, g . . y

558

ALPERIN AND GLAUBERMAN

Žb. Take x g Q. Let y s xy1 x g . Then y g Ž Q, g . and

a Ž x . s gy1 x g s x g s x xy1 x g s xy. By Ža. and Theorem 4.3, Ž Q, g . is an ideal of the Lie algebra Q. Hence, by Ž4.1., modulo this ideal we have

a Ž x . ' xy ' x q y q

1

w x, y x q ??? ' x,

2

and

Ž a y 1 . x ' a Ž x . y x ' 0. So Ž a y 1. x g Ž Q, g .. THEOREM 4.5. Assume Hypothesis 4.1. Suppose that Q has nilpotence class at most p y 1 and that R is generated by a set S of elements g such that g p centralizes A,

Ž A, g ; p . s 1,

or

Ž 4.2.

and

Ž Ž u, g ; i . , Ž ¨ , g ; j . . s 1

Ž 4.3.

for e¨ ery u, ¨ g A and all positi¨ e integers i, j such that i q j G p. Then A 1 R. Proof. Let Q0 s  x g Q < x p s 14 . Since Q has nilpotence class at most p y 1, Q is a regular p-group, whence Q0 is a subgroup of Q wHall; SII, pp. 46]47x. By Theorem 4.3, we can convert Q0 into a Lie algebra over Fp . Take any g g S, and let Q1 s ² A g < i G 0: i

and

R1 s ² A, g : .

Then Q1 is a subalgebra of Q0 . As in Lemma 4.4, let a be the automorphism of Q1 given by conjugation by g. If g p centralizes A, then

aps1

and

p

0 s Ž a p y 1 . A s Ž a y 1 . A.

Otherwise, by Ž4.2., Lemma 4.4, and an induction argument, p

Ž a y 1 . A F Ž A, g ; p . s 1. Thus in both cases, Ž a y 1. pA s 0. Let 2

Q2 s A q Ž a y 1 . A q Ž a y 1 . A q ??? q Ž a y 1 .

py1

A.

559

LIMITS OF ABELIAN SUBGROUPS

Then Q2 is invariant under a y 1, hence under a and under conjugation by g. Since A F Q2 F Q1 , we have Q2 s Q1. Therefore, Ž a y 1. p s 0. Now, from Ž4.3., we have i j Ž a y 1 . u, Ž a y 1 . ¨ s 0,

Ž 4.4.

for all u, ¨ g A and all positive integers i, j such that i q j G p. Recall the subgroups QŽ0., QŽ1., . . . , QŽ s . from Hypothesis 4.1. Now we use Ž4.4. to apply Theorem 2.8 with Q1 and A in place of L and B0 , and with L i s Q Ž i . l Q1 ,

for i s 0, 1, . . . , s.

By Hypothesis 4.1, A is maximal with respect to the partial ordering in the theorem. Therefore, A is invariant under a , that is, under conjugation by g. Since g is an arbitrary element of S, A 1 R, as desired. Now we may obtain a special case of Theorem B. COROLLARY 4.6. Suppose A 0 is an abelian subgroup of order p n in P, and P has exponent p and nilpotence class at most p y 1. Then there is a normal abelian subgroup A of P of order < A 0 < contained in ² A 0P :. Proof. Let Q s R s P. Take A as in Proposition 4.2Žb.. Let S s P. Then Theorem 4.5 applies, and yields that A 1 P. Proof of Theorem A. In Section 1, we reduced the theorem to the case where P has exponent p and order p p , and hence has class at most p y 1. Now apply Corollary 4.6. EXAMPLE 4.7 wHup, p. 349; GG1, Example 4.1; A, pp. 11]12x. Theorem A is false for n sufficiently large. The first author has given a family of examples, one for each prime p, in which < P < s p2 p

q1

2

,

< A< s p2 p

y2 pq2

2

,

and A has exactly p distinct conjugates Žincluding A. in P. Here, A is elementary abelian, and < B < - < A < for every abelian subgroup P that is not conjugate to A.

560

ALPERIN AND GLAUBERMAN

5. PROOF OF THEOREMS We have proved Theorem C in Section 3 and Theorem A in Section 4. Here, we prove the other results in our Introduction. Throughout this section, p denotes a prime and P denotes an arbitrary finite p-group. We retain the notation for commutators

Ž x, y .

Ž y, x ; i . ,

Ž Q, x .

mentioned after Theorem 4.3. From Section 1, for subgroups Q and R,

Ž Q, R . s² Ž Q, x . x g R: . For every n G 3 and for x 1 , . . . , x n g P, we define the iterated commutator Ž x 1 , . . . , x n . as usual:

Ž x 1 , x 2 , . . . , x iq1 . s Ž Ž x 1 , x 2 , . . . , x i . , x iq1 . LEMMA 5.1.

for i s 2, 3, . . . , n q 1.

Let

P s P Ž 1 . G P Ž 2 . G ??? G P Ž s . s 1 s P Ž s q 1 . s ??? be the lower central series of P, i.e., P Ž i q 1. s Ž P Ž i . , P .

for i s 1, 2, 3, . . .

Then: Ža.

For e¨ ery n G 2 and e¨ ery subset X of P that generates P, P Ž n . s² Ž x 1 , . . . , x n . , P Ž n q 1 . x 1 , . . . , x n g X: .

Žb. If P s A1 A 2 ??? A k for some abelian normal subgroups A1 , A 2 , . . . , A k of P, then P has nilpotence class at most k. Proof. Ža. See wHup, p. 258x. Žb. This is a special case of a result of Fitting Žfrom wHup, p. 276x and induction.. We will apply Lemma 5.1Žb. to a product of p normal abelian subgroups, but we wish to obtain nilpotence class at most p y 1. Therefore, we will need the following result.

561

LIMITS OF ABELIAN SUBGROUPS

LEMMA 5.2. Assume that

Suppose p G 3. A is an abelian subgroup of P, and g g P. P s ² A, g : ; g p normalizes A

A 1 ² AP :; or

Ž A, g ; p . s 1;

and

Ž Ž A, g ; i . , Ž A, g ; j . . s 1

whene¨ er i , j G 1 and i q j G p,

Then ² A P : has nilpotence class at most p y 1. Proof. Let Q s ² A P : and QX s Ž Q, Q .. Then

Ž A, ² g : . 1 ² A, ² g :: s P ,

Q s A Ž A, ² g : . ,

and

Ž 5.1. QX F Ž A, ² g : . .

Let R s ²Ž A, g; i . N i G 1:. The identities x y s x Ž x, y .

and

Ž x, y iq1 . s Ž x, y . Ž x, y i .

y

show that R g F R and, by induction, Ž A, g i . F R. Therefore, R g s R and Ž A, ² g :. F R. By Ž5.1. and the hypothesis, QX centralizes Ž A, g ; p y 1 . .

Ž 5.2.

A similar argument shows that, for each i G 0, i

² A, A g , . . . , A g : s² A, Ž A, g . , . . . , Ž A, g ; i .: . Taking i s p y 2 and i s p y 1, we obtain Q s ² A, A g , . . . , A g

py 1

: s² A, A g , . . . , A g

py 2

, Ž A, g ; p y 1 .: . Ž 5.3.

To obtain the conclusion, we must show that QŽ p . s 1. 2 py 1 By hypothesis, Q s AA g A g ??? A g . Let X be the set-theoretic union g g2 g py 1 of A, A , A , . . . , A . We will apply Lemma 5.1Žb.. Take x1 , . . . , x p g X ,

y s Ž x 1 , x 2 , . . . , x py1 . ,

and z s Ž x 1 , . . . , x p . s Ž y, x p . . We will assume that z / 1 and obtain a contradiction. r py 1y r Now x p g A g for some integer r. Replacing x i by x ig for each i, we py 1 may assume that x p g A g . Then for i s 1, 2, . . . , p y 1, x i lies in A or A g or ??? or A g

py 2

,

562

ALPERIN AND GLAUBERMAN py 1

since otherwise, x i and y lie in A g and z s Ž y, x p . s 1. Similarly, for k 1 F i - j F p y 1, x i and x j cannot lie in the same subgroup A g , 0 F k py 2 F p y 2. Therefore, y g A l A g l ??? l A g . Since p G 3, y g QX . py 2 Therefore, y centralizes A, A g , . . . , A g and Ž A, g; p y 1., by Ž5.2.. So y g ZŽ Q . by Ž5.3. and z s 1, a contradiction. It is convenient to introduce the following condition on a pair Ž Q, A. for a p-group Q and a subgroup A. HYPOTHESIS 5.3. The triple Ž Q, Q, A. satisfies Hypothesis 4.1. Now we come to our main results. We first prove an analogue of wGG2, Theorem 2Ža.x. THEOREM 5.4. Suppose A and B are elementary abelian subgroups of P, and the pairs Ž P, A. and Ž P, B . satisfy Hypothesis 5.3 Ž possibly for different central series of P .. Assume either that p s 3 and P has nilpotence class at most 4 or that p G 5. Then A and B normalize each other. Proof. We use induction on < P <. We may assume that A, B - P. Let M be a maximal subgroup of P that contains A, and let Q s ² AP :. Then Q 1 P. Since P is nilpotent, M 1 P and Q F ² M P : s M. Observe that, for each g g P, Hypothesis 5.3 is satisfied by the pair Ž P, A g . and then by the pair Ž M, A g .. Therefore, by induction, A g and Ah normalize each other for every g , h g P.

Ž 5.4.

Hence A 1 Q, and

Ž P , A . F Ž P , Q . F Q,

Ž P , A; 2 . F Ž Q, A . F A,

Ž P , A; 3 . s 1.

By the symmetry of A and B,

Ž P , B; 3 . s 1.

Ž 5.5.

Now take any u g B. Let R s ² A, u:

and

Q0 s ² A R : .

We claim that the hypothesis of Theorem 4.5 is satisfied with Q0 in place of Q and with S equal to the set-theoretic union of A with  u4 .

563

LIMITS OF ABELIAN SUBGROUPS

First of all, it is easy to see that the triple Ž Q0 , R, A. satisfies Hypothesis 4.1. If p s 3, then P has nilpotence class at most 4, whence

Ž Ž ¨ , g ; i . , Ž w, g ; j . . s 1

for all g g S, ¨ , w g A, and all i , j such that i , j G 1 and i q j G p.

Ž 5.6.

If p ) 3, we obtain Ž5.6. from Ž5.5.. Therefore, Ž5.6. holds for all p, and Lemma 5.2 yields that Q0 has nilpotence class at most p y 1. We have now verified the hypothesis of Theorem 4.5; the conclusion says that A 1 R. Since u is arbitrary in B, B normalizes A. By symmetry, A normalizes B. THEOREM 5.5. Suppose A is an elementary abelian subgroup of P and the pair Ž P, A. satisfies Hypothesis 5.3. Assume that p G 5 or that p s 3 and P has nilpotence class at most 4. Then Ža. A 1 ² A P :, and Žb. Ž P, A; 3. s 1. Proof. ŽJust repeat part of the proof of Theorem 5.4.. Now we may remove the restriction on the nilpotence class of Q in Theorem 4.5 if p G 5. THEOREM 5.6. Suppose A is an elementary abelian subgroup of P, the pair Ž P, A. satisfies Hypothesis 5.3, and p G 5 or P has exponent p. Then A is normalized by e¨ ery element g of P such that g p centralizes A,

or

Ž A, g ; p . s 1,

Ž 5.7.

and

Ž Ž u, g ; i . , Ž ¨ , g ; j . . s 1,

Ž 5.8.

for e¨ ery u, ¨ g A and all positi¨ e integers i, j such that i q j G p. Proof. If p s 2, then P is abelian. So we may assume that p G 3. Now take g g P as in the hypothesis and let R s ² A, g :

and

Q s ² AR :.

Let S be the set-theoretic union of A with  g 4 . If p s 3, then P has nilpotence class at most 3, by wHall, Ž18.2.10., p. 322x. Therefore, by Theorem 5.4, A 1 Q for all p. Then, by Lemma 5.2, Q has nilpotence class at most p y 1. Now A 1 R by Theorem 4.5. In particular, g normalizes A.

564

ALPERIN AND GLAUBERMAN

PROPOSITION 5.7. Suppose A 0 is an elementary abelian subgroup of P. Then there exists an elementary abelian subgroup A of ² A 0P : such that < A < s < A 0 < and the pair Ž P, A. satisfies Hypothesis 5.3. Proof. Take A as in Corollary 4.6, i.e., as in Proposition 4.2Žb. with Q s R s S s P. Proof of Theorem D. Here, p G 5. Apply Proposition 5.7 to obtain B, and then Theorems 5.5 and 5.6. THEOREM 5.8. Ža. Žb. Žc.

Suppose

P has nilpotence class at most p; or P has exponent p and nilpotence class at most p q 1; or p F 3 and P has exponent p.

Let A be an elementary abelian subgroup of P. Then there exists an elementary abelian subgroup B of ² A P : such that < B < s < A < and B 1 P. Remark. Consider case Žc.. From the proof of Theorem 5.6, P has class at most p. Therefore, this is a special case of Žb.. In addition, P is metabelian, so that the conclusion follows from a result of Gillam wJGx. Proof. As mentioned above, Žc. is a special case of Žb.. Now just follow the proof of Corollary 4.6, using Theorem 5.6 in place of Theorem 4.5, with slight changes. COROLLARY 5.9. Let n be a positi¨ e integer. Suppose p ) n, < P < s p n, and A is an elementary abelian subgroup of P. Then there exists an elementary abelian subgroup of ² A P : such that < B < s < A < and B 1 P. Proof of Theorem B. This is a special case of Theorem 5.8. The second author shows in wGG2x that Theorem 5.8Žc. does not extend to groups of exponent p for p ) 3. Theorem D does not apply when p - 5. The following example shows that it cannot be extended to the case in which p s 2. The case in which p s 3 is open. EXAMPLE 5.10. Suppose p s 2 and P is a dihedral group of order 32. Take x, y g P such that x 2 s y 16 s 1,

xy1 yx s yy1 ,

Let A s ² x, y 8 :. Then < A < s 4.

and

² x, y : s P.

565

LIMITS OF ABELIAN SUBGROUPS

Now A is not normal in P. Moreover, A y s ² yy1 xy, y 8 : s ² xy 2 , y 8 : and A x y s ² yy2 xy 2 , y 8 : s ² xy 4 , y 8 : / A. 2

Thus A is not normalized by A y. Similarly, for B s ² xy, y 8 :, B is not normalized by B y. However, every elementary abelian group of order 4 in P is conjugate to A or B. Thus, P has no elementary abelian subgroup E of order 4 that is normalized by P, or e¨ en by all of its conjugates in P. EXAMPLE 5.11. For P of exponent 3, P must have nilpotence class at most 3, by wHall, Ž18.2.10., p. 322x. Let P X s Ž P, P .. Then P X is abelian and 1 F Z Ž P . l PX F PX F P

Ž 5.9.

is a central series of P. For any elementary abelian subgroup A of maximal order in P, Z Ž P . l P X F Z Ž P . F A. Therefore, if we choose A such that < A l P X < is as large as possible, then Ž Q, A. satisfies Hypothesis 5.3, with Ž5.9. being the central series in condition Ž4.1.Žb.. One might guess that A G P X , since P X is abelian. But it is easy to construct a counterexample. Let F be the freest group of exponent 3 on four generators x 1 , x 2 , x 3 , x 4 . This is the group denoted by B Ž3, 4. in wHall, p. 320x. Let R be the smallest normal subgroup of F containing Ž x 1 , x 2 .. Let F s FrR ,

x i s x i R, for each i.

The analysis of F in wHall, pp. 320]324x Žparticularly Ž18.2.9.. shows that < F < s 314 ,

R s² Ž x 1 , x 2 . , Ž Ž x 1 , x 2 . , x i . i s 3, 4: ,

Ž F , F . s Ž Ž F , F . l CF Ž ² x 1 , x 2: . . =² Ž x 3 , x 4 .: ; and ² x 1 , x 2: = CŽ F , F . Ž ² x 1 , x 2: . has maximal order, which is 3 8 , while CF Ž Ž F , F . . s Ž F , F . , which has order 3 7.

< R < s 33 ;

566

ALPERIN AND GLAUBERMAN

REFERENCES wAx

J. Alperin, Large abelian subgroups of p-groups, Trans. Amer. Math. Soc. 117 Ž1965., 10]20. wBx N. Bourbaki, ‘‘Algebra,’’ Chaps. 4]7, Springer-Verlag, Berlin, 1989. wJGx J. D. Gillam, A note on finite metabelian p-groups, Proc. Amer. Math. Soc. 25 Ž1970., 189]190. wGG1x G. Glauberman, Large abelian subgroups of finite p-groups, J. Algebra 196, 301]338 Ž1997.. wGG2x G. Glauberman, Large abelian subgroups of groups of prime exponent, in preparation. wGregx T. B. Gregory, Winter map invariants, preprint. wHallx M. Hall, ‘‘The Theory of Groups,’’ Macmillan, New York, 1959. wHupx B. Huppert, ‘‘Endliche Gruppen I,’’ Springer-Verlag, Berlin, 1967. wJx N. Jacobson, ‘‘Lie Algebras,’’ Wiley, New York, 1962. wKJx M. Konvisser and D. Jonah, Counting abelian subgroups of p-groups, a projective approach, J. Algebra 34 Ž1975., 309]330. wKosx B. Kostant, Eigenvalues of a Laplacian and commutative Lie subalgebras, Topology 3, Suppl. 2 Ž1965., 147]159. wLx M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm Sup. (3) 71 Ž1954., 101]190. wSIx M. Suzuki, ‘‘Group Theory, I,’’ Springer-Verlag, Berlin, 1982. wSIIx M. Suzuki, ‘‘Group Theory, II,’’ Springer-Verlag, Berlin, 1986.