On CT and CSA Groups and Related Ideas

TL;DR

This paper explores the properties and relationships between CT (commutative transitive) and CSA (conjugately separated abelian) groups, including their implications in group theory and specific examples like PSL(2,K).

Contribution

It provides new results on the structure of finite CSA groups, characterizes when CT groups are CSA, and examines the properties of PSL(2,K) in different characteristics.

Findings

Finite CSA groups are necessarily abelian.

A CT group is not CSA iff it contains a specific nonabelian subgroup with a normal abelian subgroup.

PSL(2,K) is never CSA; it is CT under certain field characteristic conditions.

Abstract

A group is $G$ commutative transitive or CT if commuting is transitive on nontrivial elements. A group $G$ is CSA or conjugately separated abelian if maximal abelian subgroups are malnormal. These concepts have played a prominent role in the studies of fully residually free groups, limit groups and dicriminating groups. They were especially important in the solution to the Tarski problems. CSA always implies CT however the class of CSA groups is a proper subclass of the class of CT groups. For limit groups and finitely generated elementary free groups they are equivalent. In this paper we examine the relationship between the two concepts. In particular we show that a finite CSA group must be abelian. If $G$ is CT then we prove that $G$ is not CSA if and only if $G$ contains a nonabelian subgroup $G_0$ which contains a nontrivial abelian subgroup $H$ that is normal in $G_0$. For $K$ a field the group $PSL(2,K)$ is never CSA but is CT if char$(K) = 2$ and for fields $K$ of characteristic $0$ where $-1$ is not a sum of two squares in $K$. For characteristic $p$, for an odd prime $p$, $PSL(2,K)$ is never CT. Infinite CT groups $G$ with a composition series and having no nontrivial normal abelian subgroup must be monolithic with monolith a simple nonabalian CT group. Further if a group $G$ is monolithic with monolith $N$ isomorphic to $PSL(2,K)$ for a field $K$ of characteristic $2$ and $G$ is CT then $G \cong N$.