Existence, algorithms, and asymptotics of direct product decompositions, I

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 G is indecomposable, or locate a proper nontrivial direct factor. To solve this problem it is shown that in most circumstances G has a proper nontrivial subgroup M such that every maximal direct product decomposition Q of G/M induces a unique set H of subgroups of G where |H| ≤ |Q| and for each H ∈ H, the nonabelian direct factors of H are direct factors of G. In particular, G is indecomposable if |H| = 1 and M is contained in the Frattini subgroup of G. This “local-global” property of direct products can be applied inductively to M and G/M so that the existence of a proper nontrivial direct factor depends on the direct product decompositions of the chief factors of G. 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 N is reduced from the global search through all NO(log N) normal subgroups to a search of O(log N) local instances induced from chief factors. There is one family of groups G where no subgroup M admits the localglobal property just described. These are p-groups of nilpotence class 2. There are p2n 3/27+Ω(n2) isomorphism types of class 2 groups with order pn [BNV], 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 p-groups of class 2 are identified with local commutative rings. This relationship has little to do with the typical use of Lie algebras for p-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.