Glasnik Matematicki, Vol. 45, No.1 (2010), 63-83.

FINITE 2-GROUPS WITH EXACTLY ONE MAXIMAL SUBGROUP WHICH IS NEITHER ABELIAN NOR MINIMAL NONABELIAN

Zdravka Božikov and Zvonimir Janko

Mathematical Institute, University of Heidelberg, 69120 Heidelberg, Germany
e-mail: janko@mathi.uni-heidelberg.de

Abstract.   We shall determine the title groups G up to isomorphism. This solves the problem Nr.861 for p=2 stated by Y. Berkovich in [2]. The resulting groups will be presented in terms of generators and relations. We begin with the case d(G) = 2 and then we determine such groups for d(G) > 2. In these theorems we shall also describe all important characteristic subgroups so that it will be clear that groups appearing in distinct theorems are non-isomorphic. Conversely, it is easy to check that all groups given in these theorems possess exactly one maximal subgroup which is neither abelian nor minimal nonabelian.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Minimal nonabelian 2-groups, central products, metacyclic groups, Frattini subgroups, generators and relations.

DOI: 10.3336/gm.45.1.06

References:

1. Y. Berkovich, Groups of prime power order, Vol. 1, Walter de Gruyter, Berlin-New York, 2008.
   MathSciNet

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

3. Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 3, in preparation.