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Algebra Discrete Math., 2010, Volume 10, Issue 1, Pages 18–41 (Mi adm37)  

RESEARCH ARTICLE

On the existence of complements in a group to some abelian normal subgroups

Martyn R. Dixona, Leonid A. Kurdachenkob, Javier Otalc

a Department of Mathematics, University of Alabama at Tuscaloosa, AL 35487-0350, U.S.A.
b Department of Algebra, National University of Dnepropetrovsk, Dnepropetrovsk 10, 49010, Ukraine
c Departamento de Matemáticas – IUMA, Universidad de Zaragoza, 50009 Zaragoza, SPAIN

Abstract: A complement to a proper normal subgroup $H$ of a group $G$ is a subgroup $K$ such that $G=HK$ and $H\cap K=\langle 1\rangle$. Equivalently it is said that $G$ splits over $H$. In this paper we develop a theory that we call hierarchy of centralizers to obtain sufficient conditions for a group to split over a certain abelian subgroup. We apply these results to obtain an entire group-theoretical wide extension of an important result due to D. J. S. Robinson formerly shown by cohomological methods.

Keywords: Complement, splitting theorem, hierarchy of centralizers, hyperfinite group, socle of a group, socular series, section rank, $0$-rank.

Full text: PDF file (317 kB)

Bibliographic databases:
MSC: 20E22, 20E26, 20F50
Received: 02.11.2010
Revised: 02.11.2010
Language:

Citation: Martyn R. Dixon, Leonid A. Kurdachenko, Javier Otal, “On the existence of complements in a group to some abelian normal subgroups”, Algebra Discrete Math., 10:1 (2010), 18–41

Citation in format AMSBIB
\Bibitem{DixKurOta10}
\by Martyn R. Dixon, Leonid A. Kurdachenko, Javier Otal
\paper On the existence of complements in a~group to~some abelian normal subgroups
\jour Algebra Discrete Math.
\yr 2010
\vol 10
\issue 1
\pages 18--41
\mathnet{http://mi.mathnet.ru/adm37}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2807685}
\zmath{https://zbmath.org/?q=an:1212.20061}


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