2-groups of normal rank 2 for which the Frattini subgroup has rank 3

Konvisser, Marc W.

doi:10.1090/S0002-9947-1972-0292939-2

$2$-groups of normal rank $2$ for which the Frattini subgroup has rank $3$
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by Marc W. Konvisser PDF

Trans. Amer. Math. Soc. 165 (1972), 451-469 Request permission

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 ${Z_4} \times {Z_4}$, 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.

References

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