M. Arroyo-Jordá’s research while affiliated with Polytechnic University of Valencia and other places

Let π$\pi$ be a set of primes. We show that π$\pi$-separable groups have a conjugacy class of F${\mathfrak {F}}$-injectors for suitable Fitting classes F${\mathfrak {F}}$, which coincide with the usual ones when specializing to soluble groups.

Classical results from the theory of finite soluble groups state that Carter subgroups, i.e. self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble groups. This paper presents an extension of these facts to $\pi$ π -separable groups, for sets of primes $\pi$ π , by proving the existence of a conjugacy class of subgroups in $\pi$ π -separable groups, which specialize to Carter subgroups within the universe of soluble groups. The approach runs parallel to the extension of Hall theory from soluble to $\pi$ π -separable groups by Čunihin, regarding existence and properties of Hall subgroups.

Let $\pi$ be a set of primes. We show that $\pi$-separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.

Let $\pi$ be a set of primes. We show that $\pi$-separable groups have a conjugacy class of $\mathfrak F$-injectors for suitable Fitting classes $\mathfrak F$, which coincide with the usual ones when specializing to soluble groups.

Two subgroups A and B of a finite group G are said to be tcc-permutable if X permutes with for some , for all and all . Some aspects about the normal structure of a product of two tcc-permutable subgroups are analyzed. The obtained results allow to study the behaviour of such products in relation with certain classes of groups, namely the class of T-groups and some generalizations.

The study of products of groups whose factors are linked by certain permutability conditions has been the subject of fruitful investigations by a good number of authors. A particular starting point was the interest in providing criteria for products of supersoluble groups to be supersoluble. We take further previous research on total and mutual permutability by considering significant weaker permutability hypotheses. The aim of this note is to report about new progress on structural properties of factorized groups within the considered topic. As a consequence, we discuss new attainments in the framework of formation theory.

Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) if X permutes with $Y^g$ for some $g\in \langle X,Y\rangle$ , for all $X \le A$ and all $Y\le B$ . In this paper, we study finite products of tcc-permutable subgroups, focussing mainly on structural properties of such products. As an application, new achievements in the context of formation theory are obtained.

Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.

Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Yg for some element g∈G, i.e., XYg is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.