GROUPS SATISFYING THE DOUBLE CHAIN CONDITION ON ABELIAN SUBGROUPS

Abstract

If $\unicode[STIX]{x1D703}$ is a subgroup property, a group $G$ is said to satisfy the double chain condition on $\unicode[STIX]{x1D703}$-subgroups if it admits no infinite double sequences $$\begin{eqnarray}\cdots <X_{-n}<\cdots <X_{-1}<X_{0}<X_{1}<\cdots <X_{n}<\cdots\end{eqnarray}$$ consisting of $\unicode[STIX]{x1D703}$-subgroups. We describe the structure of generalised radical groups satisfying the double chain condition on abelian subgroups.