# Normality-preserving endomorphism-invariance is finite direct power-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., normality-preserving endomorphism-invariant subgroup) satisfying a subgroup metaproperty (i.e., finite direct power-closed subgroup property)

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## Statement

Suppose is a normality-preserving endomorphism-invariant subgroup of a group , i.e., for any normality-preserving endomorphism of , we have . Then, for any natural number , the subgroup is a normality-preserving endomorphism-invariant subgroup in the -fold external direct product

## Related facts

### Similar facts

- Full invariance is finite direct power-closed
- Bound-word property is finite direct power-closed
- Normal-homomorph-containment is finite direct power-closed