Existence, algorithms, and asymptotics of direct product decompositions, I
James B. Wilson · journal of Groups complexity cryptology · 2012
Direct products of finite groups are a simple method to construct new groups from old ones. A difficult problem by comparison is to prove a generic group is indecomposable, or locate a proper nontrivial direct factor. To solve this problem it is shown that in most circumstances has a proper nontrivial subgroup such that every maximal direct product decomposition of induces a unique set of subgroups of where and for each , the nonabelian direct factors of are direct factors of . In particular, is indecomposable if and is contained in the Frattini subgroup of . This “local-global” property of direct products can be applied inductively to and so that the existence of a proper nontrivial direct factor depends on the direct product decompositions of the chief factors of . Chief factors are characteristically simple groups and therefore a direct product of isomorphic simple groups. Thus a search for proper direct factors of a group of size is reduced from the global search through all normal subgroups to a search of local instances induced from chief factors. There is one family of groups where no subgroup admits the local-global property just described. These are -groups of nilpotence class 2. There are isomorphism types of class 2 groups with order , which prevents a case-by-case study. Also these groups arise in the course of the induction described above so they cannot be ignored. To identify direct factors for nilpotent groups of class 2, a functor is introduced to the category of commutative rings. The result being that indecomposable -groups of class 2 are identified with local commutative rings. This relationship has little to do with the typical use of Lie algebras for -groups and is one of the essential and unexpected components of this study. These results are the by-product of an efficient polynomial-time algorithm to prove indecomposability or locate a proper nontrivial direct factor. The theorems also explain how many isomorphism types of indecomposable groups exists of a given order and how many direct factors a group can have. These two topics are explained in a second part to this paper.