International Journal of Group Theory
Volume:4 Issue: 2, Jun 2015

It is proved here that if G is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then G is either locally supersoluble or a \vCernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.

We prove that if M is a monoid and A a finite set with more than one element, then the residual finiteness of M is equivalent to that of the monoid consisting of all cellular automata over M with alphabet A.

A subgroup H of a group G is called inert if, for each g∈G, the index of H∩H g in H is finite. We give a classification of soluble-by-finite groups G in which subnormal subgroups are inert in the cases where G has no nontrivial torsion normal subgroups or G is finitely generated.

For a composition λ of n our aim is to obtain reduced forms for all the elements in the Kazhdan-Lusztig (right) cell containing w J(λ), the longest element of the standard parabolic subgroup of S n corresponding to λ. We investigate how far this is possible to achieve by looking at elements of the form w J(λ) d, where d is a prefix of an element of minimum length in a (W J(λ), B) double coset with the trivial intersection property, B being a parabolic subgroup of S n whose type is `dual'' to that of W J(λ).

We analyze some properties of the distribution Q G,k of the first component in a k -tuple chosen uniformly in the set of all the k -tuples generating a finite group G (the limiting distribution of the product replacement algorithm). In particular, we concentrate our attention on the study of the variation distance β k (G) between Q G,k and the uniform distribution. We review some known results, analyze several examples and propose some intriguing open questions.

In this paper we illustrate recent results about factorizations of finite groups into conjugate subgroups. The illustrated results are joint works with John Cannon, Dan Levy, Attila Mar''oti and Iulian I. Simion.