On finite simple groups with a self-centralization system of type (2(𝑛))

Abstract

Abstract. Let G denote a simple group with a self-centralization system of type (2(n)), where n > 3. Let X", denote an exceptional character of G, then Xx(l) = kn + 2e where e = ± 1. It is known that \G\-^,(1X^,(1) -e)(*i+ 1) where / is a nonnegative integer. In this paper G is classified if / = 0, e = 1 and X"|(l) is odd.Let G be a finite simple group, a proper subgroup A of G is called a CC subgroup if CG(a) C A for all a E A*. If |tfc(¿)|/|¿| = 2 and \A\ = n, then G is said to have a self-centralization system of type (2(h)). The classification of simple groups with a self-centralization system of type (2(n)) is still incomplete. If n = 3, then G s PSL(2, q), q = 5 or 7 [2]. If n > 3, it is well known [5] that G has (n -l)/2 irreducible characters X" and one nonprincipal irreducible character Y such that Xx(l) = kn + 2e, Y(l) = kn + e where e = ± 1 and \G\ = nXx(l)Y(l)(ln + 1) where / is a nonnegative integer. In all the known simple groups of type (2(n)), 1 = 0 [5]. In this paper we classify all simple groups G with a self-centralization system of type (2(n)) where / = 0, e = 1, and ^,(1) is odd. In particular we prove the following: Theorem A. Let G be a finite simple group with a self-centralization system of type (2(n)) where n > 5. Let A be a subgroup of order n and let Xx be an exceptional character of G associated with N(A). Assume Xx(l) = kn + 2, ,(1) is odd and \G\ = n(kn + 2)(kn + 1), then G is isomorphic to Sz(q) or PSL(2, 2r).Let G E Hypothesis A if G satisfies the hypothesis of Theorem A but not the conclusion. Let r be an involution in N(A), Theorem 5.1 [5] implies G has one class of involutions. If S is a set, let |5| denote the number of elements in S.Assume G G Hypothesis A.If |Cc(t)| = 2r, then [6] implies either G satisfies Theorem A, G s PSL(3, 4) or G s PSL(2, q) where q is odd, n = (q + l)/2 or (q -l)/2 where n is odd. However ^,(1) is even for PSL(3, 4) or PSL(2, q), q odd. Let G2 be a Sylow 2 subgroup of G ; if G2 is abelian, then [7] again implies a

Finite groups

“…It turned out that for them IGI < 7n 5. In [687] it is proved that a simple group with a self-centralizing system of type (2n) whose order is <7n s is known (see also [685]).…”