The conjugacy problem and the inverse conjugacy problem of a finitely generated group are defined, from a language theoretic point of view, as sets of pairs of words. An automaton might be obliged to read the two input words synchronously, or could have the option to read asynchronously. Hence each class of languages gives rise to four classes of groups; groups whose (inverse) conjugacy problem...

In this paper we present a new, practical algorithm for solving the subgroup conjugacy problem in the general linear group. Mathematics Subject Classification: 20-04, 20H30. 1 Introductory Material This paper presents a new algorithm to solve a subcase of the following: Problem. Given two groups G,H ≤ K, determine whether there exists a k ∈ K such that G = H. If so, return one such k. This prob...

Let G ⊂ SL(n, C) be a finite subgroup and φ:Y → X = C/G any resolution of singularities of the quotient space. We prove that crepant exceptional prime divisors of Y correspond one-to-one with “junior” conjugacy classes of G. When n = 2 this is a version of the McKay correspondence (with irreducible representations of G replaced by conjugacy classes). In the case n = 3, a resolution with KY = 0 ...

In [8, §8], the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow p-subgroup U(q) of a finite Chevalley group G(q), valid when q is a power of a good prime for G(q). In this paper we develop this algorithm and discuss an implementation in the computer algebra language GAP. Using the resulting computer program we are able to calculate the pa...

(A) We study in this paper topological properties of conjugacy classes in Polish groups. There are two questions which we are particularly interested in. First, does a Polish group G have a dense conjugacy class? This is equivalent (see, e.g., Kechris [95, 8.47]) to the following generic ergodicity property of G: Every conjugacy invariant subset A ⊆ G with the Baire property (e.g., a Borel set)...

Let G be the group of k-points of a connected reductive k-group and H a symmetric subgroup associated to an involution σ of G. We prove a polar decomposition G = KAH for the symmetric space G/H over any local field k of characteristic not 2. Here K is a compact subset of G and A is a finite union of groups Ai which are the k-points of maximal (k, σ)-split tori, one for each H-conjugacy class. T...

Definitions and classifications of reduced surds for non-arithmetical groups on the hyperbolic plane is proposed, as well those some congruence subgroups. The considered are desymmetrized group, its Γ0 subgroup, a generalized Hecke group according choice commutator subgroup group. definitions studied to ordering generators conjugacy subclasses groups. construction tori defined obtained after te...

Let G be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in G of any element of G. Let H be a subgroup of G. We give some conditions on H , under which we provide a solution to the conjugacy problem in H . We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy probl...