Twisted conjugacy classes of the unit element

Abstract

In this paper we study twisted conjugacy classes of the unit element in different groups. A. L. Fel'shtyn and E. V. Troitsky showed that in an abelian group a twisted conjugacy class of the unit element is a subgroup for any automorphism. In this article we study the question about a structure of the groups, where twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).

On groups where the twisted conjugacy class of the unit element is a subgroup

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We study groups G where the ϕ-conjugacy class [e] ϕ = {g −1 ϕ(g) | g ∈ G} of the unit element is a subgroup of G for every automorphism ϕ of G. If G has n generators, then we prove that the k-th member of the lower central series has a finite verbal width bounded in terms of n, k. Moreover, we prove that if such group G satisfies the descending chain condition for normal subgroups, then G is nilpotent, what generalizes the result from [4]. Finally, if G is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of G.

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Twisted Conjugacy Classes in Chevalley Groups

“…In this section we state some sufficient conditions when the twisted conjugacy class of the unit element is a subgroup of Chevalley group. At first, let us remind three propositions from the paper [11].…”

“…This class contains unit element, whence the following question arises: for which groups G and its automoprhisms ϕ class [e] ϕ is a subgroup of group G? In the paper [11] it is stated that for automorphism ϕ, which acts identically modulo center of group G, class [e] ϕ is a subgroup of G. In present paper we prove that for Chevalley groups over some fields this result is a criterion, i. e. if G is a Chevalley group over the field F which is determined in the theorem 1 or in the theorem 2, than the ϕ-conjugacy class of the unit element [e] ϕ is a subgroup of G if and only if ϕ acts identically modulo center of G (theorems 3 and 4).…”

Automorphism groups of quandles arising from groups

“…This set is the so called ϕ-twisted conjugacy class of the identity element and is of independent interest. See [2] for some related results. When G is a finite group, this condition is equivalent to ϕ being fixed-point free.…”