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By Steinke G. F.

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F(X) n S (F(X) n S)(E(X) n S) =: Proof. First F(X) r S Then by induction F(S) ~ F(S) F(X) n S. Then C =: ~ F(S)E(S). =: F(S). ) for all i 1 =: S

For generalizations of the techniques of the following sections, the reader should consult Goldschmidt's 'strongly closed abelian' paper [11]. 12. PRELIMINARY LEMMAS Lemma 12. 1 (Thompson). Sylow 2-subgroup S. r(R) = r(S) - L Let G be a group with an abelian Let R be a subgroup of S such that If G = 02(G), then any involution t E S is conjugate in NG(S) to an element of R. Proof. Of course, NG(S) controls fusion of elements of S by Burnside's lemma. Consider the transfer V: G -+ S. Let Xl' ... , Xk ' ~+ 1 t , ...

J. H. Waiter obtained a characterization of finite groups with abelian Sylow 2-subgroups in [22], [23]. Bender offers a novel approach in [5]. In this section we commence this classification by Bender of groups with abelian Sylow 2-subgroups. This depends on the character- ization also due to Bender of groups which have a strongly embedded subgroup. , There are many equivalent formulations of this concept. The following definition suffices for our purposes here. Definition H. 1. A subgroup H of even order of a group G is X strongly embedded in G if H *- G and H n H has odd order for all xEG-H.

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