A homological characterization of locally finite groups. (English) Zbl 1255.20045
Let \(G\) be a group. We say that \(G\) is locally finite if every subgroup of it that is finitely generated, is finite. We define the generalized homological dimension \(\text{hd}(G)\) (resp. the modified generalized homological dimension \(\text{hd}'(G)\)) as the supremum of those integers \(n\), for which there exist a \(\mathbb Z\)-torsion-free (resp. \(\mathbb Z\)-free) \(\mathbb ZG\)-module \(M\) and an injective \(\mathbb ZG\)-module \(I\), such that \(\text{Tor}_n^{\mathbb ZG}(M,I)\neq 0\).
In this paper the author proves that the locally finiteness of \(G\) is equivalent to \(\text{hd}(G)=0\) and to \(\text{hd}'(G)=0\).
In this paper the author proves that the locally finiteness of \(G\) is equivalent to \(\text{hd}(G)=0\) and to \(\text{hd}'(G)=0\).
Reviewer: Saïd Zarati (Tunis)
MSC:
| 20J05 | Homological methods in group theory |
| 20F50 | Periodic groups; locally finite groups |
| 16E10 | Homological dimension in associative algebras |
| 16S34 | Group rings |
| 18G20 | Homological dimension (category-theoretic aspects) |
| 20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
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