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Groupprops β

Directly indecomposable group

Definition

A nontrivial group is said to be directly indecomposable if it satisfies the following equivalent conditions:

Formalisms

In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: direct factor
View other properties obtained by applying the simple group operator

The group property of being directly indecomposable is obtained by applying the simple group operator to the subgroup property of being a direct factor.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
simple group nontrivial and no proper nontrivial normal subgroup Group having no proper nontrivial transitively normal subgroup, Monolithic group, Splitting-simple group, Subdirectly irreducible group|FULL LIST, MORE INFO
subdirectly irreducible group nontrivial and cannot be expressed as a nontrivial subdirect product |FULL LIST, MORE INFO
splitting-simple group nontrivial and no proper nontrivial complemented normal subgroup |FULL LIST, MORE INFO
centrally indecomposable group nontrivial and any central factor is inside the center |FULL LIST, MORE INFO
quasisimple group perfect and its inner automorphism group is simple Splitting-simple group|FULL LIST, MORE INFO
almost simple group has a normal fully normalized subgroup that is simple non-abelian |FULL LIST, MORE INFO

Stronger properties conditional to nontriviality

The following properties are stronger, assuming the group is nontrivial.

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which every endomorphism is trivial or an automorphism any endomorphism is either the trivial map or an automorphism Splitting-simple group|FULL LIST, MORE INFO

Facts

Products of directly indecomposable groups

It is clear that every finite group can be expressed as a product of directly indecomposable groups. The question: is this expression as a product essentially unique? That is, is there an analogue of unique factorization for direct products? The answer is yes, as per the Remak-Schmidt theorem.