engleski

Finite 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian

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 (Theorems 2.1, 2.2 and 2.3) and then we determine such groups for d(G) > 2 (Theorems 3.1, 3.2 and 3.3). 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.

minimal nonabelian 2-groups; central products; metacyclic groups; Frattini subgroups; generators and relations

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