6533b873fe1ef96bd12d5596

RESEARCH PRODUCT

On groups with abelian Sylow 2-subgroups

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Finite groups with abelian Sylow 2-subgroups have been classified by Walter [8]. In this note I want to describe an alternate proof of some partial result of Walter's work, namely the theorem stated below. It represents the first major reduction step in that classification. The approach used here is to some extent derived from [1]. ! Besides the groups L 2 (q)= PSL(2, q) another class of simple groups enters our discussion: We say that a simple group G with abelian Sz-subgroups is of type JR (Janko-Ree) if, for any involution t in G, CG (t) is a maximal subgroup of G isomorphic to ( t ) | E where PSL(2, q)~ E ~_ PFL(2, q) with odd q > 5. In fact, E = L 2 (q), as proved by Walter 1-7] ; and the structure of G is very well known by results of Janko, Thompson, and Ward, see [4-6] and [9]. Unpublished work of Thompson is very close to a complete identification of such a group G. For further information about the subject see the introduction of 1-8] and 1-3; 16.6]. A group G with abelian S2-subgroups is an A*-group if G has a normal series 1 _~ N_~ M _~ G such that N and G/M are (solvable) of odd order and M/N is a direct product of a 2-group and simple groups of type L 2 (q) or JR. Note that the result of Walter mentioned above means that an A*-group is an A-group (in the sense of 1-3; 16.6]). The rank r(X) of a group X is defined in the following way: X has an elementary abelian p-subgroup of order p,(X), for a suitable prime p, but none of order prtX)+1, for any prime p. Throughout this paper, "g roup" means "finite group".