A note on co-Hopfian groups and rings

TL;DR

This paper investigates conditions under which special linear groups over certain fields are co-Hopfian, extending previous results, and shows that Turner groups are not closed under elementary equivalence.

Contribution

It proves that SL_2 over the algebraic closure of the field with two elements is co-Hopfian, extending known results and addressing a question about Turner groups.

Findings

SL_2 over the algebraic closure of the field with two elements is co-Hopfian.

The class of Turner groups is not closed under elementary equivalence.

The result extends the understanding of co-Hopfian properties in algebraic groups.

Abstract

Let $p$ and $n$ be positive integers. Assume additionally that $p\neq 3$ is a prime and that $n>2$. Let $R$ be a field of characteristic $p$. A very special consequence of a result of Bunina and Kunyavskii (2023, arXiv:2308.10076) is that $SL_{n}(R)$ is co-Hopfian as a group if and only if $R$ is co-Hopfian as a ring. In this paper, we prove that if $k$ is the algebraic closure of the $2$ element field, then $SL_{2}(k)$ is a co-Hopfian group. Since this $k$ is trivially seen to be co-Hopfian as a ring our result somewhat extends that of Bunina and Kunyavskii. We apply our result to prove that the class of groups satisfying Turner's Retract Theorem (called Turner groups here) is not closed under elementary equivalence thereby answering a question posed by the authors in (2017, Comm. Algebra).