Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N . We show that the hyperbolic group from F. Haglund’s and D. Wise’s version of Rips’s construction is hereditarily conjugacy separable. We then use this construction to produce firs...

We evaluate one-dimensional representations of quantum symmetric conjugacy classes classical matrix groups along with their stabilizer subgroups.

Let G be a finite group and C(G) be the family of representative conjugacy classes of‎ ‎subgroups of G‎. ‎The matrix whose H,K-entry is the number of ‎fixed points of the set G/K under the action of H is called the‎ ‎table of marks of G where H,K run through all elements in‎ C(G)‎. Shinsaku Fujita for the first time introduced the term “markaracter” to discuss marks for permutation representati...

We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a longstanding open problem. The proof uses, among other tools, some methods from Geometric Invariant Theory. Using this result we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of L...

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results...

We announce our successful computation of a list of representatives of the conjugacy classes of subgroups of Sn for n ≤ 18, including the 7 274 651 classes of subgroups of S18.