in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

We describe generators and defining relations for the commutator subgroup of topological full groups of minimal subshifts. We show that the world problem in a topological full group is solvable if and only if the language of the underlying subshift is recursive.

Let G be a group. We shall denote by L(G) the lattice formed by all subgroups of G. Two groups G and H are said to be lattice-isomorphic, or in short P-isomorphic, when their lattices L(G) and L(H) are isomorphic to each other. In this case the isomorphism from L(G) onto L(H) is called the L-isomorphism from G onto H. If G and H are isomorphic as groups, then G is clearly P-isomorphic to H. The...

We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a “hybrid group” approach; that is, we first compute a large solvable normal subgroup of the given permutation group and then use this to split the computation in various parts. 1991 Mathematics Subject Classification: primary 20B40, 20-04, 20E28; secondary 20B15, 68Q40

We introduce a combinatorial version of Stallings-Bestvina-Feighn-Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.

Let G be a finite almost simple group with socle G0. A (nontrivial) factorization of is an expression the form G=HK, where factors H and K are core-free subgroups. There extensive literature on factorizations groups, important applications in permutation theory algebraic graph theory. In recent paper, Xia second-named author describe groups solvable factor H. Several infinite families arise con...

A well known unresolved conjecture states that every Cayley graph on a solvable group G has a 1-factorization. We show that if the commutator subgroup of of such a group is an elementary abelian p-group, then every quartic Cayley graph on G has a 1-factorization.

We review some aspects of gauged WZW models. By choosing a solvable subgroup as gauge group, one is lead to three main applications: the construction of field theories with an extended conformal symmetry, the construction of the effective action of (extended) 2D gravities and the systematic construction of string theories with some extended gauge symmetry.

‎For a homogeneous spaces ‎$‎G/H‎$‎, we show that the convolution on $L^1(G/H)$ is the same as convolution on $L^1(K)$, where $G$ is semidirect product of a closed subgroup $H$ and a normal subgroup $K $ of ‎$‎G‎$‎. ‎Also we prove that there exists a one to one correspondence between nondegenerat $ast$-representations of $L^1(G/H)$ and representations of ...

We show that if G1 and G2 are non-solvable groups, then no C1,τ action of $(G_{1}\times G_{2})*\mathbb {Z}$ on S1 is faithful for τ > 0. As a corollary, S an orientable surface complexity at least three the critical regularity arbitrary finite index subgroup mapping class group Mod(S) with respect to circle most one, thus strengthening result first two authors Baik.