A descent principle in modular subgroup arithmetic

Proceedings of the American Mathematical Society, 1969

Journal of Algebra

Given a p-group G and a subgroup-closed class X, we associate with each X-subgroup H certain quantities which count X-subgroups containing H subject to further properties. We show in Theorem I that each one of the said quantities is always ≡ 1 (mod p) if and only if the same holds for the others. In Theorem II we supplement the above result by focusing on normal X-subgroups and in Theorem III we obtain a sharpened version of a celebrated theorem of Burnside relative to the class of abelian groups of bounded exponent. Various other corollaries are also presented.

Journal of Algebra, 2004

Let p be a prime and for a finite p-group G let 0 (0) (G) = G and 0 (i) (G) = 0 1 (0 (i−1) (G)) for i ∈ N. A theorem is proved stating that, if exp(G) = q > and cl(G) = c, then the length of the 0 (i)series cannot be bounded by a function of q alone; an upper bound in terms of q and c is given which is shown to be attained in p-groups of arbitrarily large class.

This note deals with the computation of the factorization number F 2 (G) of a finite group G. By using the Möbius inversion formula, explicit expressions of F 2 (G) are obtained for two classes of finite abelian groups, improving the results of Factorization numbers of some finite groups, Glasgow Math. J. (2012).

Proceedings of the London Mathematical Society, 1976

Journal of Algebra, 1975

H generated by their p-subgroups, and thereby deduce bounds for their defects in terms of other invariants of a Sylow p-subgroup P. Writing, respectively, 3P) and Kr for each integer Y to mean the rth term of the derived series of a group K and the subgroup generated by the rth powers of the elements of K, we may state 242 Copyright

Monatshefte für Mathematik, 2020

Let the group G = AB be the product of subgroups A and B, and let p be a prime. We prove that p does not divide the conjugacy class size (index) of each p-regular element of prime power order x ∈ A ∪ B if and only if G is p-decomposable, i.e. G = O p (G) × O p ′ (G).

Journal of Group Theory, 2009

Given a group G and subgroups X d Y , with Y of finite index in X , then in general it is not possible to determine the index jX : Y j simply from the lattice 'ðGÞ of subgroups of G. For example, this is the case when G has prime order. The purpose of this work is twofold. First we show that in any group, if the indices jX : Y j are determined for all cyclic subgroups X , then they are determined for all subgroups X. Second we show that if G is a group with an ascending normal series with factors locally finite or abelian, and if the Hirsch length of G is at least 3, then all indices jX : Y j are determined. The first author acknowledges support from GNSAGA and hospitality from the University of Padova. The second author acknowledges hospitality from the University of Warwick.

Proceedings of the London Mathematical Society, 1964

To appear in Advanced Research in Pure Mathematics