Über minimale k-irreguläre p-Gruppen. II. (On minimally k-irregular p-groups. II)
This paper is a continuation of the author's previous work [Part I, ibid. 24, 111-123 (1987; Zbl 0618.20014)] on minimally k-irregular p-groups, and begins with two generalizations of criteria of Mann for regularity. Next a p-group is defined to be k-V-regular if all the finite groups in the variety generated by it are k-regular, and criteria for this are given. Conditions for a p-group to be \(p^ k\)-Abelian are then given, and it is shown that the variety generated by any minimally k-irregular p-group can be generated by a cyclic and a minimally k-irregular p-group of exponent \(p^{k+1}\). Next p-groups which are minimally k-irregular in the sense of subgroups are studied and generalizations of Mann's theorems are given. Finally it is shown that if P is a minimally k-irregular p- group of class \(c\geq p\) and exponent \(p^ n\) \((n>k)\), then there exists a p-group M minimally k-irregular in the sense of subgroups with the properties that M is of exponent \(p^ n\) and class c, the exponent of M' is at most \(p^ k\), P is a homomorphic image of M and M is isomorphic to a subgroup of the direct product of three certain quotient groups of it.
scientific article; zbMATH DE number 4004447
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