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We prove that the solvable radical of a finite group G coincides with the set of elements y
having the following property: for any x∈ G the subgroup of G generated by x and y is …

We present a survey of results on word equations in simple groups, as well as their
analogues and generalizations, which were obtained over the past decade using various …

We obtain the following characterization of the solvable radical R (G) of any finite group G: R
(G) coincides with the collection of all g∈ G such that for any 3 elements a1, a2, a3∈ G the …

We prove that an element g of prime order> 3 belongs to the solvable radical R (G) of a finite
(or, more generally, a linear) group if and only if for every x∈ G the subgroup generated by …

For a variety of finite groups H, let H‾ denote the variety of finite semigroups all of whose
subgroups lie in H. We give a characterization of the subsets of a finite semigroup that are …

We study dynamical systems arising from word maps on simple groups. We develop a
geometric method based on the classical trace map for investigating periodic points of such …

The solvable Farb growth of a group quantifies how well-approximated the group is by its
finite solvable quotients. In this note we present a new characterization of polycyclic groups …

We introduce Extended special linear group $ ESL_2 (k) $, which is generalization of matrix
group $ SL_2 (k) $ over perfect field $ k $. Extended special linear group $ ESL_2 (k) …

We give a survey of new characterizations of finite solvable groups and the solvable radical
of an arbitrary finite group which were obtained over the past decade. We also discuss …