Directly indecomposable group

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


 * It has no proper nontrivial direct factor
 * It cannot be expressed as an external direct product of nontrivial groups

Definition with symbols
A group $$G$$ is said to be directly indecomposable if we cannot write $$G$$ as:

$$G \cong H \times K$$

with both $$H$$ and $$K$$ being nontrivial groups.

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

Stronger properties

 * Weaker than::Simple group
 * Weaker than::Subdirectly irreducible group
 * Weaker than::Splitting-simple group
 * Weaker than::Centrally indecomposable group
 * Weaker than::Group in which every endomorphism is trivial or an automorphism
 * Weaker than::Quasisimple group

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.