Existence, algorithms, and asymptotics of direct product decompositions, I
-
James B. Wilson
Abstract.
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.
© 2012 by Walter de Gruyter Berlin Boston
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- Continuous hard-to-invert functions and biometric authentication
- Existence, algorithms, and asymptotics of direct product decompositions, I
- Isomorphism in expanding families of indistinguishable groups
- Search and test algorithms for triple product property triples
- Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography
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Articles in the same Issue
- Masthead
- Two-party key establishment: From passive to active security without introducing new assumptions
- Continuous hard-to-invert functions and biometric authentication
- Existence, algorithms, and asymptotics of direct product decompositions, I
- Isomorphism in expanding families of indistinguishable groups
- Search and test algorithms for triple product property triples
- Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography
- A Diffie–Hellman key exchange protocol using matrices over noncommutative rings
- No-leak authentication by the Sherlock Holmes method
