Glasnik Matematicki, Vol. 46, No.1 (2011), 71-77.

ON FINITE p-GROUPS CONTAINING A MAXIMAL ELEMENTARY ABELIAN SUBGROUP OF ORDER p²

Yakov Berkovich

Abstract.   We continue investigation of a p-group G containing a maximal elementary abelian subgroup R of order p², p>2, initiated by Glauberman and Mazza [6]; case p=2 also considered. We study the structure of the centralizer of R in G. This reduces the investigation of the structure of G to results of Blackburn and Janko (see references). Minimal nonabelian subgroups play important role in proofs of Theorems 2 and 5.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Minimal nonabelian p-group, maximal elementary abelian subgroup, soft subgroup.

DOI: 10.3336/gm.46.1.09

References:

1. Y. Berkovich, Groups of prime power order. Vol. 1, Walter de Gruyter, Berlin, 2008.
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2. Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 2, Walter de Gruyter, Berlin, 2008.
   MathSciNet    

3. Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 3, Walter de Gruyter, Berlin, 2011.

4. N. Blackburn, Generalizations of certain elementary theorems on p-groups, Proc. London Math. Soc. (3) 11 (1961), 1-22.
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6. G. Glauberman and N. Mazza, p-groups with maximal elementary abelian subgroup of rank 2, J. Algebra 323 (2010), 1729-1737.
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7. L. Hethelyi, Some remarks on 2-groups having soft subgroups, Studia Sci. Math. Hungar. 27 (1992), 295-299.
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8. Z. Janko, Finite 2-groups with small centralizer of an involution, J. Algebra 241 (2001), 818-826.
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9. Z. Janko, Finite 2-groups with small centralizer of an involution, 2, J. Algebra 245 (2001), 413-429.
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