Geodesic
On infinite Alperin groups
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 210-222

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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
Mots-clés : torsion-free group. 
@article{SEMR_2015_12_a6,
     author = {B. M. Veretennikov},
     title = {On infinite {Alperin} groups},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {210--222},
     publisher = {mathdoc},
     volume = {12},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2015_12_a6/}
}
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B. M. Veretennikov. On infinite Alperin groups. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 210-222. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a6/