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...

We give another proof of an observation of Thévenaz [4] and present a fusion system version of it. Namely, for a saturated fusion system F on a finite p-group S, we show that the number of the F-conjugacy classes of cyclic subgroups of S is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realiz...

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem”. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a ...

Suppose $G$ is a connected complex semisimple group and $W$ its Weyl group. The lifting of an element to semisimple. This induces well-defined map from the set elliptic conjugacy classes $G$. In this paper, we give uniform algorithm compute map. We also consider twisted case.

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group G, Parry showed how to define a G-extension SA from a square matrix over Z+G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over Z+G. Parry asked in this case if the dynamical zeta fun...

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...

In this short note, the conjugacy classes of finite dihedral subgroups of the 4 × 4 integral symplectic group are considered. A complete list of representatives of the classes is obtained, among them six classes are realizable by analytic automorphisms of compact connected Riemann surfaces of genus two.

Abstract We prove that there exists a universal constant D such if p is prime divisor of the index Fitting subgroup finite group G , then number conjugacy classes at least $Dp/\log_2p$ . conjecture we can take $D=1$ and for solvable groups, $D=1/3$