Geodesic
On the existence of complements in a~group to~some abelian normal subgroups
Algebra and discrete mathematics, Tome 10 (2010) no. 1, pp. 18-41.

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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: splitting theorem, hierarchy of centralizers, hyperfinite group, socular series, section rank, $0$-rank.
Mots-clés : Complement, socle of a group 
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Martyn R. Dixon; Leonid A. Kurdachenko; Javier Otal. On the existence of complements in a~group to~some abelian normal subgroups. Algebra and discrete mathematics, Tome 10 (2010) no. 1, pp. 18-41. http://geodesic.mathdoc.fr/item/ADM_2010_10_1_a3/