Cohopfian groups and accessible group classes

Francesco de Giovanni and Marco Trombetti

Vol. 312 (2021), No. 2, 457–475

DOI: 10.2140/pjm.2021.312.457

Abstract

A group $G$ is said to be cohopfian if it is neither trivial nor isomorphic to any of its proper subgroups, and this property is equivalent to the existence of a suitable group class $\mathfrak{𝔛}$ such that $G$ is minimal non-$\mathfrak{𝔛}$. If $\mathfrak{𝔛}$ is any group class, the subclass ${\mathfrak{𝔛}}^{\circ }$ consisting of all groups that are isomorphic to proper subgroups of locally graded minimal non-$\mathfrak{𝔛}$ groups is often much smaller than $\mathfrak{𝔛}$. Similarly, if ${\mathfrak{𝔛}}^{prop}$ is the class of all groups isomorphic to proper subgroups of $\mathfrak{𝔛}$-groups, the class $\overline{\mathfrak{𝔛}}$ of all locally graded minimal non-${\mathfrak{𝔛}}^{prop}$ groups may contain many groups which are not in $\mathfrak{𝔛}$. This paper investigates the relation between the classes $\mathfrak{𝔛}$, ${\mathfrak{𝔛}}^{\circ }$ and $\overline{\mathfrak{𝔛}}$.

Keywords

cohopfian group, cohopfian group class, accessible group class

Mathematical Subject Classification 2010

Primary: 20E34

Milestones

Received: 26 February 2020

Revised: 7 January 2021

Accepted: 3 May 2021

Published: 31 August 2021