-groups of normal rank for which the Frattini subgroup has rank
Abstract: All finite 2-groups G with the following property are classified: Property. The Frattini subgroup of G contains an abelian subgroup of rank 3, but G contains no normal abelian subgroup of rank 3.
The method of classification involves showing that if G is such a group, then G contains a normal abelian subgroup W isomorphic to , and that the centralizer C of W in G has an uncomplicated structure. The groups with the above property are then constructed as extensions of C.
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-  B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
-  Anne R. MacWilliams, On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups, Trans. Amer. Math. Soc. 150 (1970), 345–408. MR 0276324, https://doi.org/10.1090/S0002-9947-1970-0276324-3
- J. Alperin, Centralizers of Abelian normal subgroups of p-groups, J. Algebra 1 (1964), 110-113. MR 29 #4800. MR 0167528 (29:4800)
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, Berlin, 1967. MR 0224703 (37:302)
- A. R. MacWilliams, On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups, Trans. Amer. Math. Soc. 153 (1970), 345-408. MR 0276324 (43:2071)
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Keywords: 2-group, embedding of abelian subgroups, normal abelian subgroup, involution, normal abelian subgroup of rank 3
Article copyright: © Copyright 1972 American Mathematical Society