Abstract:
All non-abelian finite simple groups can be symmetrically generated by involutions. An algorithm which performs coset enumeration for a group defined in this manner on the cosets of a subgroup of automorphisms of these involutions is presented.

Abstract:
The following problem is proposed as Problem 18.57 in [The Kourovka Notebook, No. 18, 2014] by D. V. Lytkina:\\ Let $G$ be a finite $2$-group generated by involutions in which $[x, u, u] = 1$ for every $x \in G$ and every involution $u \in G$. Is the derived length of $G$ bounded?\\ The question is asked of an upper bound on the solvability length of finite $2$-groups generated by involutions in which every involution (not only the generators) is also left $2$-Engel. We negatively answer the question.

Abstract:
In this paper, we show that $\sigma$-involutions associated to extendable c=4/5 Virasoro vectors generate a 3-transposition group in the automorphism group of a vertex operator algebra (VOA). Several explicit examples related to lattice VOA are also discussed in details. In particular, we show that the automorphism group of the VOA $V_{K_{12}}^{\hat{\nu}}$ associated to the Coxeter Todd lattice $K_{12}$ contains a subgroup isomorphic to ${}^+\Omega^{-}(8,3)$.

Abstract:
A double-coset enumeration algorithm for groups generated by symmetric sets of involutions together with its computer implementation is described.

Abstract:
We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n_F_(n-1) are thin.

Abstract:
Let $\Sigma_{g,b}$ denote a closed oriented surface genus $g$ with $b$ punctures and let $Mod_{g,b}$ denote its mapping class group. Luo proved that if the genus is at least 3, the group $Mod_{g,b}$ is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate $Mod_{g,b}$. Brendle and Farb gave a partial answer in the case of closed surfaces and surfaces with one puncture, by describing a generating set consisting of 7 involutions. Our main result generalizes the above result to the case of multiple punctures. We also show that the mapping class group can be generated by smaller number of involutions. More precisely, we prove that the mapping class group can be generated by 4 involutions if the genus $g$ is large enough. There is not a lot room to improve this bound because to generate this group we need at lest 3 involutions. In the case of small genus (but at least 3) to generate the whole mapping class group we need a few more involutions.

Abstract:
We discuss basic structural properties of finite black box groups. A special emphasis is made on the use of centralisers of involutions in probabilistic recognition of black box groups. In particular, we suggest an algorithm for finding the $p$-core of a black box group of odd characteristic. This special role of involutions suggest that the theory of black box groups reproduces, at a non-deterministic level, some important features of the classification of finite simple groups.

Abstract:
It is shown that a finite group in which more than 3/4 of the elements are involutions must be an elementary abelian 2-group. A group in which exactly 3/4 of the elements are involutions is characterized as the direct product of the dihedral group of order 8 with an elementary abelian 2-group.

Abstract:
We associate a 2-complex to the following data: a presentation of a semigroup $S$ and a transitive action of $S$ on a set $V$ by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Sch\"utzenberger groups. We also give a geometric proof that a finitely generated regular semigroup with finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtually nilpotent.