# Full invariance is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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## Statement

### Statement with symbols

Suppose $H \le K \le G$ are groups (in words, $H$ is a subgroup of $K$ and $K$ is a subgroup of $G$). Then, if $K$ is a fully invariant subgroup of $G$ and $H$ is a fully invariant subgroup of $K$, we have that $H$ is a fully invariant subgroup of $G$.

## Facts used

1. Balanced implies transitive

## Proof

### Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic

Full invariance is the balanced subgroup property for endomorphisms. In other words, it can be expressed as:

endomorphism $\to$ endomorphism

This says that $K$ is fully invaraint in $G$ if every endomorphism of $G$ restricts to an endomorphism of $K$. By fact (1), any balanced subgroup property is transitive, hence full invariance is transitive.