On generalised FC-groups in which normality is a transitive relation
We extend to soluble FC∗ -groups, the class of generalised FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes , Serdica Math. J. 28(3) (2002), 241 254], the characterisation of finite soluble T-groups obtained recently in [G. Kaplan, On T-groups, supersolvable groups and maximal subgroups , Arch. Math. 96 (2011), 19 25].
On finite p-groups of supersoluble type
Abstract A finite p-group S is said to be of supersoluble type if every fusion system over S is supersoluble. The main aim of this paper is to characterise the finite p-groups of supersoluble type. Abelian and metacyclic p-groups of supersoluble type are completely described. Furthermore, we show that the Sylow p-subgroups of supersoluble type of a finite simple group must be cyclic.
Algorithms for permutability in finite groups
In this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP.
A question from the Kourovka Notebook on formation products
[EN] It is shown in this paper that if X is a class of simple groups such that pi(X) = char X, the X-saturated formation H generated by a finite group cannot be expressed as the Gaschütz product F o G of two non-X-saturated formations if H = G. It answers some open questions on products of formations. The relation between omega-saturated and X-saturated formations is also discussed.
Maximal subgroups of small index of finite almost simple groups
We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.
Left braces and the quantum Yang-Baxter equation
[EN] Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang¿Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang¿Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang¿Baxter equation.