On finite 2-groups with non-Dedekind norm of abelian non-cyclic subgroups

F. M. Lyman; T. D. Lukashova; M. G. Drushlyak

doi:10.15330/ms.46.1.20-28

The authors study finite 2-groups with the cyclic center and non-metacyclic non-Dedekind norm of Abelian non-cyclic subgroups. It is found out that such groups are cyclic or metacyclic extensions of their norms of Abelian non-cyclic subgroups. Their structure is described.

1. R. Baer, Der Kern eine charakteristische Untergruppe, Comp. Math., 1 (1935), 254-283.

2. T.D. Lukashova, Locally finite $p$-groups ($p\ne 2$) with non-Abelian norm of non-cyclic subroups, Bulletin of Kiev Univ., 1 (2001), 43-53.

3. T.D. Lukashova, On the norm of Abelian non-cyclic subgroups in infinite locally finite $p$-groups ($p\ne 2$), Bulletin of Kiev Univ., 3 (2004), 35-39.

4. F.M. Lyman, T.D. Lukashova, On infinite 2-groups with non-Dedekind norm of Abelian non-cyclic subgroups, Bulletin of Kiev Univ., 1 (2005), 56-64.

5. M.G. Drushlyak, Finite $p$-groups ($p\ne 2$) with non-Abelian norm of Abelian non-cyclic subgroups, Proceedings of Francisk Scorina Gomel State Univ., 1 (2010), 192-197.

6. F.M. Lyman, T.D. Lukashova, M.G. Drushlyak, Finite 2-groups with non-Dedekind norm of Abelian non-cyclic subgroups and non-cyclic center, Bulletin of Kiev Univ., 1 (2012), 26-32.

7. F.M. Lyman, p-groups, in which all Abelian non-cyclic subgroups are invariant, Dopovidi AN USSR, 8 (1968), 696-699.

8. M. Hall, Group Theory, Izdatelstvo inostrannoy literatury, Moscow, 1962. (in Russian)

9. V.A. Sheriev, Finite 2-groups with complemented non-invariant subgroups, Siberian Math. Journal, 8 (1967), 195-213.

	On finite 2-groups with non-Dedekind norm of abelian non-cyclic subgroups
Author	F. M. Lyman, T. D. Lukashova, M. G. Drushlyak mathematicsspu@mail.ru Makarenko Sumy State Pedagogical University
Abstract	The authors study finite 2-groups with the cyclic center and non-metacyclic non-Dedekind norm of Abelian non-cyclic subgroups. It is found out that such groups are cyclic or metacyclic extensions of their norms of Abelian non-cyclic subgroups. Their structure is described.
Keywords	group; non-Dedekind group; non-metacyclic group; norm of group; norm of Abelian non-cyclic subroups
DOI	doi:10.15330/ms.46.1.20-28
Reference	1. R. Baer, Der Kern eine charakteristische Untergruppe, Comp. Math., 1 (1935), 254-283. 2. T.D. Lukashova, Locally finite $p$-groups ($p\ne 2$) with non-Abelian norm of non-cyclic subroups, Bulletin of Kiev Univ., 1 (2001), 43-53. 3. T.D. Lukashova, On the norm of Abelian non-cyclic subgroups in infinite locally finite $p$-groups ($p\ne 2$), Bulletin of Kiev Univ., 3 (2004), 35-39. 4. F.M. Lyman, T.D. Lukashova, On infinite 2-groups with non-Dedekind norm of Abelian non-cyclic subgroups, Bulletin of Kiev Univ., 1 (2005), 56-64. 5. M.G. Drushlyak, Finite $p$-groups ($p\ne 2$) with non-Abelian norm of Abelian non-cyclic subgroups, Proceedings of Francisk Scorina Gomel State Univ., 1 (2010), 192-197. 6. F.M. Lyman, T.D. Lukashova, M.G. Drushlyak, Finite 2-groups with non-Dedekind norm of Abelian non-cyclic subgroups and non-cyclic center, Bulletin of Kiev Univ., 1 (2012), 26-32. 7. F.M. Lyman, p-groups, in which all Abelian non-cyclic subgroups are invariant, Dopovidi AN USSR, 8 (1968), 696-699. 8. M. Hall, Group Theory, Izdatelstvo inostrannoy literatury, Moscow, 1962. (in Russian) 9. V.A. Sheriev, Finite 2-groups with complemented non-invariant subgroups, Siberian Math. Journal, 8 (1967), 195-213.
Pages	20-28
Volume	46
Issue	1
Year	2016
Journal	Matematychni Studii
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