manuscripta math. 70, 255 - 260 (1991) manuscripta mathematica �9 S~inger-Verlag 1991

R E G U L A R O R B I T S O F H A L L 7 r - S U B G R O U P S

By Alberto Espuelas and Gabriel Navarro

1. I n t r o d u c t i o n

Let V be a faithful KG-module and let H be a subgroup of G. When

does V contain a regular H-orbit?

If H is a p-subgroup of the solvable group G and Op(G) = 1, it has

been proved by the first author that H always has a regular orbit on V,

being p = char(K) odd ([1]).

This (not semisimple) result had Hall-Higman type applications : if

G is a solvable group and p is an odd prime number, then IG : Op,p(G)lp

divides b(P), where P is a Sylow-p-subgroup of G and (as in [4]) b(P) =

max{x(1) [ X �9 Irr(P)}.

It is not in general true that any subgroup H of G has a regular orbit

on V. However, if O,~(G) = 1, where ~r is the set of primes which divide

]HI, we will show that H does have a regular orbit on V (when the group

G is of odd order).

It is our aim to prove the following.

T h e o r e m . Let G be a group of odd order and let H be a Hall ~r-

subgroup of G. Let V be a faithful G-module, over possibly different finite

~elds of odd 7r-characteristic. Assume that Vo,(c) is completely reducible.

Then there exists v �9 V such that CH(V) C_ O,~(G).

As in [1], we give the following character degree application.

C o r o l l a r y . Let G be a group of odd order and let H be a Hall ~r-

subgroup of G. Then there exists a �9 Irr(H) such that [G : O,m~(G)[,~

divides ce(1). /n particular, [G : O,,,~(G)[,~ _< b(H) .

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2. P r o o f s

Next L e m m a is a powerful tool for looking for regular orbits.

L e m m a . Let G be a group of odd order having a faithful and irre-

ducible quasiprimitive module V over a finite ~eld of odd characteristic.

Suppose that F(G) is noncyclic. Then V contains at least two regular

G-orbits.

Proof. See Lemma 2.1. of [2].

P r o o f o f t h e T h e o r e m . Let G be a counterexample minimizing

dimK(g).

(1) V is G-irreducible.

Proof. Let R be a Hall m-subgroup of O,~,~,(G). If h E H -O,,(G), let

1 # Y(h) a Hall 7r/-subgroup or [h, R].

If h E H-O,~(G), we claim that there exists an irreducible G-submodule

V(h) of V such tha t Y(h) acts nontrivially on V(h).

Since Vo.(a) is completely reducible and the fields have r-characteris t ic

observe tha t Vo,.,(a) is also completely reducible.

Write Vo,.,(a) = V1 @ . . . @ V,, where the ~ ' s are the homogeneous

components.

Since Y(h) > 1, suppose for instance that Y(h) acts nontrivially on

Vl.

Now consider the G-submodule ~ , e a VlX and choose an irreducible

G-submodule W of it. We claim that V1 N W > 0. To prove this, let X be

an irreducible O,~,~,(a)-submodule of W. Since for every x E G, the V/x ' s

are homogeneous components, it follows that X C_ Vlx , for some x E G.

Since Wx = W, we will have tha t 1/'1 N W > 0.

Suppose now that Y(h) acts trivially on W fl V1 and let Y be an

irreducible O,~,(a)-submodule of W N V1. Therefore, since V1 is a direct

sum of modules isomorphic to Y, it follows that Y(h) acts trivially on 1/1.

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This shows that Y(h) acts nontrivially on V(h) = W, as claimed.

If H = O,~(G) there is nothing to prove. Let U = ~he~-O, (a) V(h)

a completely reducible G-submodule of V. If U < V, by induction, there

exists u E U such that Co(u) C O,~(G), where G = G/Ca(U). Let C =

ca(u). We claim that H N C = O~( G).

Let h E H n C - Or(G). Since h E C, it follows that h acts trivially

on V(h). Therefore, [h,R] C_ kerV(h) and thus Y(h) acts trivially on

V(h), which is a contradiction. This proves that H N C C O~(G). Now,

H N C C_ On(G) n C = O~(C) C H n C, as claimed.

Now we prove that O~(G) = O~(G)C/C. Let K / C = O~(G).

Observe that [K/COw(G), O~,(G)/CO~(G)] = 1.

If h E H n K - O~(G), then [h,R] C_ [K,O~,(G)] C O~(G)C. Since

C contains the rr'-subgroups of O~(G)C, it follows that Y(h) C C, which

is a contradiction. This proves that K = O~(G)C , as we wanted.

Now, since H N C = O,~(G) and CH(U) C 07r(G)C, it follows that

CH(U) C__ O,~(G) and we may assume that U = V.

Hence V is a completely reducible G-module.

If V is not irreducible, V = V1 @ V2, where each 1~ is G-invariant. By

induction, let vi E ~ such that CHCG(VI)/CG(VI)(Vi) ~ O~r(G/Ca(Yi)) and

consider v = vl + v2. Then Cg(v) C_ On(G).

Thus, we may assume that V is an irreducible KG-module, for a finite

field K with odd characteristic.

(2) V is a quasiprimitive G-module.

Proof. Let N be a normal subgroup of G and put VN = V1 @ ... @ Vt,

where the I,~'s are homogeneous N-modules. Moreover, ~ is an irreducible

Na(Vi)-module with V/a = V. Let H1 be a Hall 7r-subgroup of Na(V1)

containing NH(V1) and suppose that t > 1.

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By induction, there exists v 6 V1 such that CH, CG(Vl)/c6(v~)(Vl) C_

O.(NaCV,)/CaW,)) = SlC (V, ).

Now, G acts transitively on fl = {Vl , . . . ,Vt} with kernel Ge.

By Corollary 1 of [3], we have that G / G ~ has a regular orbit on the

power set of ~t. Let A C_ ~ be a representative of such an orbit. Let gi C G

with V~gi = Vi.

Let w = xl + . . - + xf E V be defined as follows: xi = vgi if ~ E A and

xi = - v g i i f Vi 6 f~ - A.

Observe that since G is a group of odd order and the characteristic of

the field is odd, ux # - u for all z E G and u 6 V.

We see that CH(W) C O,~(G). Suppose that h E CH(w) and write

Vjh = Va0-).Then x i h = xa(j). Since there is no x E G with vx = - v , we

have that Va(j) E A if P~ E A. This implies that h 6 G~ (i.e., a( j ) = j for MI

j ) . Consequently, gjhg-i -1 6 C~%(v,)(v) for all j . Hence h E A~=I ..... t S~i,

which is a normal 7r-subgroup of G.

(3) F ( G ) is cyclic.

Proof. If not, Lemma above gives us the existence of a regular orbit.

By Proposition (2.1) of [5], we may assume that V = GF(q") + and

that G C J, where J = C , I~ M, (M = GF(q'~) * acts as multiplications on

V and C , = G a l ( G r ( q " ) / G F ( q ) ) ) . Moreover, G M M = F(G) .

(4) There is no counterexample.

Proof. Let K and < cr > be Hall T-subgroups of M and C , , respec-

tively, and let G* = < a > t<K. Replacing G by some conjugate we may

assume that G C_ G*.

Also, observe that if the theorem is true for G* it is also true for G.

Hence, it is not loss to suppose G = G*.

Replacing again by some conjugate, we may assume that H = < ~- >

~<K=, where < r > and K,~ are Hall ~r-subgroups of < cr > and K, respec-

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tively.

It suffices to show that [UheI~-O~(G) Cv(h)[ < [V[.

Write H - On(G) = Ui=l ..... ~ x jK~, where xf e < r >. Suppose that

UheH--O.(e) Cv(h) = V. Then

v # = U j = , ..... ,(U..,e~-. c, , , ( .~ j , , ' , ) ) .

Since for each j, Ume,% C v # ( x j m ) = {t~ ~_ V # I ,,~" /~, e K, , } i~ a

multiplicative subgroup of g # (cyclic), it follows that

V # -- UmeK. C v # ( x j m ) , for some j .

FolIowing notation and Proposition (1.3) of [5I, if N = {x e V # [Nzj (z) -

1), where N~ is the norm map (i.e., N~(y) = y a ( y ) . . , a~ then n _ 1

]N[-- ~ , where s = o(xj).

By Proposition (1.3) of [5], we will have that N C K~. Then it follows

that [K~,[ divides qn/, _ 1.

Since, by Galois Theory, ICv#(xj)[ = q~/" - 1, we have that gr C_

Cv#(x~). Thus < xj,K,~ > < < x j , K > ~ G.

Since xj r Or this is a contradiction.

Coro l la ry . Let G be a group of odd order and Iet H be a Hall ~r-

subgroup of G. Then there exists a E lrr(H) such that tG : O,,~(G)f,~

aiviaes ~(1). I~ p ~ i c ~ r , fC : O.,,(G)I, _< b(H).

Proof. We may assume that O,r,(G) -- 1. Let N = O~(G). Then,

fairly standard arguments show that C --- C G ( F ( N ) / ~ ( N ) ) c Y . Write

V = I r r ( F ( N ) / ~ ( N ) ) and 0 = G/C. Thus, On(G) = N / C .

Now, V is a faithful G-module such that Vo. (c ) is completely reducible.

By the Theorem, there exists A e Irr(V) such that C~(A) C N. Let ~ E

Irr(CH(A)[~) and a = ~a e Irr(H). Thus ]H: N[ divides a(1), as wanted.

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References

1. A. Espuelas, The Existence of regular orbits, J. Algebra 127, No.

2, (1989), 259-268

2. A. EspueIas, Large character degrees of groups of odd order, To

appear in Illinois J. Math.

3. D. Gluck, Trivial set stabilizers in finite permutation groups, Can.

J. Math. 35, (1983), 59-67

4. L M. Iseacs, Character Theory of Finite Groups, New York, Aca-

demic Press, 1976

5. A. Turu/1, Supersotvable automorphism groups of solvable groups,

Math. Z.183, (1983), 47-73

Departamento de Algebra. Facultad de Ciencias.Universidad de Zaragoza.

Zaragoza. Spain.

Departamento de Algebra. Facultad de Ciencias Matematicas. Uni-

versitat de Valencia. Burjassot. Valencia. Spain.

Research partially supported by DGICYT

(Received June 11, 1990; in r e v i s e d form December 19, 1990)