Groups of prime power order. Vol. 5.
A few years ago the reviewer was pleasantly surprized with the announcement that after the successfull three volumes regarding explicit structure of \(p\)-groups by \textit{Y. G. Berkovich} [Groups of prime power order. Vol. 1. de Gruyter Expositions in Mathematics 46. Berlin: Walter de Gruyter (2008; Zbl 1168.20001)] and \textit{Y. G. Berkovich} and \textit{Z. Janko} [Groups of prime power order. Vol. 2. de Gruyter Expositions in Mathematics 47. Berlin: Walter de Gruyter (2008; Zbl 1168.20002); Groups of prime power order. Vol. 3. de Gruyter Expositions in Mathematics 56. Berlin: Walter de Gruyter (2011; Zbl 1229.20001)], another two volumes were in progress to appear. Seeing over all five volumes now, one is able to conclude that the authors have enriched the mathematical literature with an important topic. The two books here under review do contain a wealth of new results. To give a ``numerical'' impression firstly: In volume 4 that are at least 24 new characaterizations of certain classes of \(p\)-groups together with 46 sections filled with new structure theorems; 13 so-called Appendices, hundreds of exercises and 508 (unsolved mostly) problems and research themes. In volume 5 these figures are respectively (in short) 34; 66; 52; hundreds; 572. All these things constitute a remarkable treasure cove for researchers and other curious mathematiciens in the years to come. Here are some snapshorts to be found in volume 5: (partly quoted from backover) \(\bullet\) \(p\)-groups containing a subgroup of maximal class of index \(p\); \(\bullet\) normal closures of nonnormal subgroups; \(\bullet\) \(p\)-groups all of whose subgroups are isomorphic to quotient groups and \(p\)-groups all of whose quotient groups are isomorphic to subgroups; \(\bullet\) \(p\)-groups with many minimal nonabelian subgroups; \(\bullet\) generalizations of Dedekindian groups; \(\bullet\) \(p\)-groups all of whose nonnormal subgroups have exponent \(p\); \(\bullet\) structure of \(p\)-groups with cyclic derived subgroup for odd \(p\); \(\bullet\) meta-Hamiltonian \(p\)-groups; \(\bullet\) \(p\)-groups that are not generated by their nonnormal subgroups; \(\bullet\) \(p\)-groups saturated by isolated subgroups or otherwise by nonabelian Dedekind subgroups; \(\bullet\) \(p\)-groups with an Abelian subgroup of index \(p\).
scientific article; zbMATH DE number 6532320