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- Title
On maximal abelian subgroups in finite 2-groups.
- Authors
Zvonimir Janko
- Abstract
Abstract  We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups (Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation) about the structure of finite nonabelian p-groups G such that A ⩠B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).
- Subjects
ISOMORPHISM (Mathematics); NONABELIAN groups; ABELIAN groups; MATHEMATICS
- Publication
Mathematische Zeitschrift, 2008, Vol 258, Issue 3, p629
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-007-0189-1