# Subdirectly irreducible group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definitionTemplate:NotaproductVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Subdirectly irreducible group, all facts related to Subdirectly irreducible group) |Survey articles about this | Survey articles about definitions built on this

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## Definition

### Symbol-free definition

A group is said to be **subdirectly irreducible** if:

- Any expression of the group as subdirect product has that the projection map to at least one of the factors is an isomorphism
- The trivial subgroup of the group cannot be expressed as an intersection of two nontrivial normal subgroups