This article is cited in 1 paper

Mathematical logic, algebra and number theory

On infinite Alperin groups

B. M. Veretennikov

Ural Federal University, Ekaterinburg, Mira 19, 620002, Ekaterinburg, Russia

Abstract: A group $G$ is called Alperin group if any 2-generated subgroup of $G$ has a cyclic commutator subgroup. We prove the existence of Alperin torsion-free groups and Alperin groups, generated by involutions, with free abelian second commutator subgroups of any finite and countable rank. Also we prove that nilpotent torsion-free Alperin group has nilpotence class $\leq 2$. The last theorem of the article implies that the following condition is insufficient for a group $G$ to be Alperin group:
$$\text{for any } a,b \in G \text{ commutator } [a,b,b] \text{ is a power of } [a,b].$$


Keywords: Alperin group, commutator subgroup, generators and defining relations, Hopfian group, torsion-free group.

UDC: 512.54

MSC: 20B05

Received February 18, 2015, published March 20, 2015

DOI: 10.17377/semi.2015.12.017