Regular subgroups of the affine group and asymmetric product of radical braces. (English) Zbl 1348.20002

Catino, Francesco; Colazzo, Ilaria; Stefanelli, Paola

J. Algebra 455, 164-182 (2016).

M. W. Liebeck et al. [Mem. Am. Math. Soc. 952 (2010; Zbl 1198.20002)] asked for a classification of all pairs \((G,B)\) where \(G\) is a finite permutation group and \(B\) a regular subgroup. An interesting class of examples for the affine case was given by P. Hegedűs [J. Algebra 225, No. 2, 740-742 (2000; Zbl 0953.20040)]. The problem to classify regular subgroups of an affine group is equivalent to the classification of braces [F. Catino and R. Rizzo, Bull. Aust. Math. Soc. 79, No. 1, 103-107 (2009; Zbl 1184.20001)].
In the paper under review, the authors introduce the concept of asymmetric product of braces, a semidirect product twisted by a 2-cocycle. This sheds some light on Hegedűs’ construction.

Reviewer: Wolfgang Rump (Stuttgart)

MSC:

20B15	Primitive groups
16N20	Jacobson radical, quasimultiplication

Keywords:

finite permutation groups; affine groups; regular subgroups; braces; asymmetric products

Citations:

Zbl 1198.20002; Zbl 0953.20040; Zbl 1184.20001

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