Related papers: Conjugation in Semigroups

We consider two relations on a $\cap$-semigroup of partial functions of a given set: the inclusion of domains and the semiadjacencity (i.e., the inclusion of the image of the first function into the domain of the second), which…

Various kinds of infinitary operations satisfying forms of associativity have been considered in the literature by various authors, including A. Tarski, C. Karp, J. H. Conway, D. Krob, N. Bedon, and C. Rispal. Applications include the…

We consider finite groups having a conjugacy class that is the difference of two normal subgroups. That is, suppose $G$ is a group and $M$ and $N$ are normal subgroups so that $N < M$, and suppose that there is an element $g \in G$ so that…

In this note, we determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion $2$-groups. A generalization of this property is also…

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental…

We study conjugacy relations on semigroups and monoids, focusing on the relation $a \cfn b$, defined by the existence of $g,h \in S^1$ such that $ag = gb$, $bh = ha$, $hag = b$, and $gbh = a$. This notion emerged as one that yields…

In this paper, we count the number of conjugacy and automorphism-conjugacy classes of elements of a ZM-group. The size of a conjugacy class with respect to these two equivalence relations is also counted.

We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation…

We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.

We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. As all previously known examples, our examples are contracted semigroup algebras and the underlying…

New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.

We characterize the group property of being with infinite conjugacy classes (or icc, i.e. in which all conjugacy classes beside 1 are infinite) for split extensions of groups.

This survey article explores the notion of z-classes in groups. The concept introduced here is related to the notion of orbit types in transformation groups, and types or genus in the representation theory of finite groups of Lie type. Two…

The paper begins by exploring the various definitions of norms on semigroups and then presents a new definition of a normed semigroup. The properties of normed semigroups in the new sense are investigated. The new definition of the norm is…

Given a semigroup $S$ and $s,t \in S$, write $s \sim_p^1 t$ if $s=pr$ and $t=rp$, for some $p,r \in S \cup \{1\}$. This relation, known as "primary conjugacy", along with its transitive closure $\sim_p$, has been extensively used and…

Let $G$ be a finite group. Let $k(G)$ denote the number of conjugacy classes of $G$ and let $m(G)$ denote the least positive integer $n$ such that the union of any $n$ distinct non-trivial conjugacy classes of $G$ together with the identity…

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider…

The study of rational relations is fundamental to the study of formal languages and automata theory. A rational relation is conjugate if each pair of words in the relation is conjugate (or cyclic shifts of each other). The notion of…

Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set…

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field…