ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 4, Pages 88–90 (November 10, 1998) S 1079-6762(98)00051-1

NONABELIAN SYLOW SUBGROUPS OF FINITE GROUPS OF EVEN ORDER

NAOKI CHIGIRA, NOBUO IIYORI, AND HIROYOSHI YAMAKI

(Communicated by Efim Zelmanov)

Abstract. We have been able to prove that every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution. The proof depends upon the consequences of the classification of finite simple groups.

The purpose of this note is to announce [4]:

Main Theorem. Every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution.

Let G be a finite group and Γ(G) the prime graph of G. Γ(G) is the graph such that the vertex set is the set of prime divisors of |G|, and two distinct vertices p and r are joined by an edge if and only if there exists an element of order pr in G. Let n(Γ(G)) be the number of connected components of Γ(G) and d_{G}(p, r) the distance between two vertices p and r of Γ(G). It has been proved that n(Γ(G)) ≤ 6 in [12], [10], [11], [9].

Theorem 1. Let G be a finite group of even order and p a prime divisor of |G|. If d_{G}(2, p) ≥ 2, then a Sylow p-subgroup of G is abelian.

Theorem 1 is a restatement of the Main Theorem in terms of the prime graph Γ(G).

Corollary 1. Let G be a finite group of even order and p a prime divisor of |G|. If Δ is a connected component of Γ(G)−{p} not containing 2, then a Sylow r-subgroup of G is abelian for any r ∈ Δ.

There is a certain relation between a subgraph Γ(G) − {p} of Γ(G) and Brauer characters of p-modular representations of G (see [3]).

Theorem 2. Let G be a finite nonabelian simple group and p an odd prime divisor of |G|. Then d_{G}(2, p) = 1 or 2 provided d_{G}(2, p) < ∞.

The significance of the prime graphs of finite groups can be found in [1], [3], [5], [6], [7], [8], [14], [15]. We apply the classification of finite simple groups (see [1], [2], [4], [7], [10], [13]). It has been proved that a minimal counterexample to Theorem 1

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors October 20, 1997.

1991 Mathematics Subject Classification. Primary 20D05, 20D06, 20D20.

Key words and phrases. Sylow subgroups, prime graphs, simple groups.

The third author was supported in part by Grant-in-Aid for Scientific Research (No. 8304003, No. 08640051), Ministry of Education, Science, Sports and Culture, Japan.

is a nonabelian simple group. We will give some examples of case by case analysis for finite simple groups. Theorem 1 holds true for the sporadic simple groups by Atlas of Finite Groups although we can find several typos in it. For a positive integer k let π(k) be the set of all prime divisors of k. Let π_{0} = {p ∈ π(G) | d_{G}(2, p) = 1}. Then we do not have to think about primes in π_{0} in order to give the proof of Theorem 1.

Example. Let G be the alternating group on n-letters and p ∈ π(G). It is trivial that Theorem 1 holds true for A_{5} and A_{6}. Assume that n ≥ 7. If p ≤ n − 4, then d_{G}(2, p) = 1. If p ≥ n − 3, then Sylow p-subgroups of G are cyclic of order p. Thus Theorem 1 holds true for the alternating groups.

Example. Let G = P SL(n, q), q ≡ 0 (2). Then |G| = q^{n(n−1)/2} ∏_{i=1}^{n−1}(q^{i+1} − 1)d^{−1},

d = (n, q − 1). Let I_{j} be the j × j identity matrix. Put

t′_{k} = (\begin{matrix}I_{k} & 0 & 0 \\ 0 & I_{n−2k} & 0 \\ I_{k} & 0 & I_{k} \end{matrix}) .

Then t′_{k} (1 ≤ k ≤ r), where r = [n/2], are representatives of the conjugacy classes of involutions in SL(n, q). The centralizer of t′_{k} in SL(n, q) is the set of all matrices of the form

(\begin{matrix}A & 0 & 0 \\ H & B & 0 \\ K & L & A \end{matrix}) ,

where (det A)^{2} det B = 1 and A is a k × k nonsingular matrix. Denote t_{k} the homomorphic image of t′_{k} in P SL(n, q). Then t_{k} (1 ≤ k ≤ r) are representatives of the conjugacy classes of involutions in P SL(n, q). Let C_{k} = C_{G}(t_{k}). Then

π(C_{k}) = π(2 ∏_{i=1}^{n−2k}(q^{i} − 1)/(q − 1)d)

and

π_{0} = π(∏_{k=1}^{r} |C_{k}|) = π(2 ∏_{i=1}^{n−2}(q^{i} − 1)).

Suppose n ≥ 4. Then the only factors of |G| to be considered are (q^{n−1} −1)(q^{n} −1). There are maximal tori T (A_{n−2}) of order (q^{n−1} − 1)d^{−1} and T (A_{n−1}) of order (q^{n} − 1)/(q − 1)d. Let p ∈ π(T (X)) − π_{0}, where X = A_{n−1} or A_{n−2}. Let P be a Sylow p-subgroup of T (X). Then d_{G}(2, p) = 1 or P is a Sylow p-subgroup of G. Since P is abelian, Theorem 1 holds true for G = P SL(n, q), n ≥ 4.

Suppose that n = 3. Then |G| = q^{3}(q^{2} −1)(q^{3} −1)d^{−1} and there are three classes of maximal tori of orders

(q − 1)^{2}d^{−1}, (q^{2} − 1)d^{−1}, (q^{2} + q + 1)d^{−1}.

We note that a torus of order (q^{2} + q + 1)d^{−1} is an isolated subgroup. If q > 4, then d_{G}(2, r) = 2 for r ∈ π(q + 1). Let R be a Sylow r-subgroup of G. Then R is contained in a maximal torus of order (q^{2} − 1)d^{−1}. If q = 4, then G = P SL(3, 4) and |G| = 2^{6} · 3^{2} · 5 · 7. If q = 2, then G = P SL(3, 2) and |G| = 2^{3} · 3 · 7. We have verified Theorem 1 for n = 3. It is trivial that Theorem 1 holds true for P SL(2, q).

Theorem 3. Let G be a simple group of Lie type and T a maximal torus. Let p ∈ π(T) − π_{0}. Then T contains a Sylow p-subgroup of G.

Theorem 3 is a corollary of Theorem 1. Actually we prove Theorem 3 for specified tori when we verify Theorem 1 for the simple groups of Lie type.

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Department of Mathematical Sciences, Muroran Institute of Technology, Hokkaido 050-8585, Japan

E-mail address: chigira@muroran-it.ac.jp

Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi 753-8512, Japan

E-mail address: iiyori@po.yb.cc.yamaguchi-u.ac.jp

Department of Mathematics, Kumamoto University, Kumamoto 860-8555, Japan

E-mail address: yamaki@gpo.kumamoto-u.ac.jp