Glasnik Matematicki, Vol. 40, No.1 (2005), 71-86.

FINITE 2-GROUPS G WITH |Ω2(G)| = 16

Zvonimir Janko

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

Abstract.   It is a known fact that the subgroup Ω2(G) generated by all elements of order at most 4 in a finite 2-group G has a strong influence on the structure of the whole group. Here we determine finite 2-groups G with |G| > 16 and Ω2(G) = 16. The resulting groups are only in one case metacyclic and we get in addition eight infinite classes of non-metacyclic 2-groups and one exceptional group of order 25. All non-metacyclic 2-groups will be given in terms of generators and relations.

In addition we determine completely finite 2-groups G which possess exactly one abelian subgroup of type (4,2).

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   2-group, metacyclic group, Frattini subgroup, self-centralizing subgroup.

Full text (PDF) (free access)

DOI: 10.3336/gm.40.1.08

References:

  1. Y. Berkovich, On subgroups and epimorphic images of finite p-groups, J. Algebra 248 (2002), 472-553.

  2. Y. Berkovich, Selected topics of finite group theory, in preparation.

  3. Z. Janko, Finite 2-groups with small centralizer of an involution, 2, J. Algebra, 245 (2001), 413-429.
    CrossRef

  4. Z. Janko, Finite 2-groups with no normal elementary abelian subgroups of order 8, J. Algebra 246 (2001), 951-961.
    CrossRef

  5. Z. Janko, Finite 2-groups with a self-centralizing elementary abelian subgroup of order 8, J. Algebra 269 (2003), 189-214.
    CrossRef
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