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SPECIAL MAXIMAL SUBGROUPS OF $p$-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  07 August 2013

JOHN COSSEY
JOHN COSSEY*
Affiliation:
Department of Mathematics, Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia email [email protected]
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Abstract

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In the 2006 edition of the Kourovka Notebook, Berkovich poses the following problem (Problem 16.13): Let $p$ be a prime and $P$ be a finite $p$-group. Can $P$ have every maximal subgroup special? We show that the structure of such groups is very restricted, but for all primes there are groups of arbitrarily large size with this property.

Keywords

p-groupsspecial maximal subgroups

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 415 - 419
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Beisiegel, B., ‘Semi-extraspezielle $p$-Gruppen’, Math. Z. 156 (1977), 247–254.CrossRefGoogle Scholar
Doerk, K. and Hawkes, T., Finite Soluble Groups, De Gruyter Expositions in Mathematics, 4 (De Gruyter, Berlin–New York, 1992).Google Scholar
Heineken, H., ‘Engelsche Elemente der Länge drei’, Illinois J. Math. 5 (1961), 681–707.CrossRefGoogle Scholar
Huppert, B., Endliche Gruppen I (Springer, Berlin, Heidelberg, New York, 1967).Google Scholar
Mazurov, V. D. and Khukhro, E. I. (eds.) Unsolved Problems in Group Theory: The Kourovka Notebook, 16th edn (Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 2006).Google Scholar
Neumann, H., Varieties of Groups (Springer, Berlin, Heidelberg, New York, 1967).Google Scholar