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On the number of commutators in groups. (English) Zbl 1122.20017
Arad, Zvi (ed.) et al., Ischia group theory 2004. Proceedings of a conference in honor of Marcel Herzog, Naples, Italy, March 31–April 03, 2004. Providence, RI: American Mathematical Society (AMS); Ramat Gan: Bar-Ilan University (ISBN 0-8218-3711-7/pbk). Contemporary Mathematics 402. Israel Mathematical Conference Proceedings, 181-192 (2006).
This interesting article is dedicated to the groups with a finite set of subgroups serving as commutator subgroups of some subgroups of the group (C-groups). The authors obtain a general description of C-groups and some details of the structure of perfect C-groups with no proper subgroups of finite index. Among others, they prove that a C-group has finitely generated commutator subgroup. The authors also describe the groups in which every non-Abelian subgroup has the commutator subgroup coinciding with the commutator subgroup of the whole group. In particular, they prove that such a group is solvable if and only if it is locally graded.
For the entire collection see [Zbl 1089.20500].

MSC:
20F14 Derived series, central series, and generalizations for groups
20E34 General structure theorems for groups
20F16 Solvable groups, supersolvable groups
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
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