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Let T be any finite 2-group which has a normal four-group but has no normal Abelian subgroup of rank 3, and assume T is not the dihedral group of order 8. If T is a Sylow 2-subgroup of a finite fusion-simple group G, it follows (Thompson) from Glauberman's Z*-theorem that T has exactly one normal four-group, say W. This paper establishes what isomorphism types of T can so occur under the hypothesis that H^CT) = TCAT) and the three nonidentity elements of W are not all G-conjugate. All T arrived at in this paper are known to so occur. The reason for this hypothesis is that the similar situation for T with a normal four-group and no normal Abelian subgroup of rank 3, where T is a Sylow 2-subgroup of a finite simple group G but without the above hypothesis, had been analyzed earlier by the author (under her maiden name, MacWilliams; Trans. Amer. Math. Soc. 150 (1970), 345-408). The result of Feit and Thompson [7] suggests that one might classify finite simple groups by their Sylow 2-subgroups. One measure of the complexity of the possible 2-structure of a group is its normal 2-rank, i.e., the largest rank of an Abelian 2-subgroup which is normal in some Sylow 2-subgroup containing it. If the normal 2-rank is 1, then the Sylow 2-subgroup is cyclic or of maximal class (Blackburn [3]), and the simple groups with normal 2-rank 1 are known (Brauer, Gorenstein, Walter, Alperin [4], [9], [2]). The present paper is a completion of [12], in which a start was made on finding the 2-groups T that can occur as Sylow 2-subgroups of simple groups G with normal 2-rank 2; except that in the present paper we actually assume only that G is fusion-simple, i.e., that G has no proper 2-quotient, no nonidentity normal subgroup of odd order, and no nonidentity central elements. We use the following result of Thompson, of which a proof is given in [12]: If G is a fusion-simple group with normal 2-rank 2, then the Sylow 2-subgroups Received by the editors September 11, 1972. AMS (MOS) subject classifications (1970). Primary 20D05, 20D20; Secondary 20D15.

On Sylow 2-subgroups with no normal Abelian subgroups of rank 3, in finite fusion-simple groups