A decomposition theory for finite groups with applications to 𝑝-groups

Abstract

Introduction. This note introduces a notion of decomposition for groups which includes direct product and subdirect product as special cases. First a closure operator on sets of groups is introduced which embeds a given set of groups I Ga} into its closure, denoted by { G,} I*, which is the smallest set of groups containing {Ga } and closed under the operations of taking direct products, subgroups and factor groups. A G is then called an in-direct product if GE {Aa} * where {Aa} is the set of all proper subgroups and factor groups of G. Some necessary conditions for decomposability are derived and it is shown, in particular, that an in-direct product which is not a subdirect product or a factor of a direct product contains a normal subgroup which is an abelian p-group. In ?2 an attempt is made to gain information about the decomposability of a p-group G from the knowledge that G/Z(G) is an in-direct product. It is shown that whenever the class of G is greater than two then G is an indirect product if G/Z(G) is a factor of a direct product. An example is given which shows that this decomposition is in general nontrivial. That is, an in-direct product is given which is neither a subdirect product, or the factor of a direct product. In ?3 several characterizations of those p-groups of class two which are not in-direct products are given. The following notation will be used throughout without special definition: A CB, A is a proper subgroup (subset) of B. (a,, * , a.) is the generated by the set { a,, ... , a.} . The direct product A XB-{ [a, b I I a EA, b EB }. o(A) is the order of the (element) A. Z(G) is the center of G. Zj(G) is defined by: Zj(G)/Zj_j(G) = Z(G/Z_1(G)) for i =2, 3, with Z(G) =Z1(G). c(G) is the class of G. From this point on the word group will mean finite group. 1. Since it will be convenient to refer interchangeably to the notions of subgroup, factor and factor of a subgroup we state: DEFINITION 1.1. A A is said to be an ingroup of a G if one of the following holds: