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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Marc W. Konvisser
Journal: Trans. Amer. Math. Soc. 165 (1972), 451-469
MSC: Primary 20D15
DOI: https://doi.org/10.1090/S0002-9947-1972-0292939-2
MathSciNet review: 0292939
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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 [Enhancements On Off] (What's this?)

  • [1] J. Alperin, Centralizers of Abelian normal subgroups of p-groups, J. Algebra 1 (1964), 110-113. MR 29 #4800. MR 0167528 (29:4800)
  • [2] B. Huppert, Endliche Gruppen. I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, Berlin, 1967. MR 0224703 (37:302)
  • [3] 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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0292939-2
Keywords: 2-group, embedding of abelian subgroups, normal abelian subgroup, involution, normal abelian subgroup of rank 3
Article copyright: © Copyright 1972 American Mathematical Society

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