# Full invariance is transitive

From Groupprops

(Redirected from Full characteristicity 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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Get more facts about fully invariant subgroup |Get facts that use property satisfaction of fully invariant subgroup | Get facts that use property satisfaction of fully invariant subgroup|Get more facts about transitive subgroup property

## Contents

## Statement

### Statement with symbols

Suppose are groups (in words, is a subgroup of and is a subgroup of ). Then, if is a fully invariant subgroup of and is a fully invariant subgroup of , we have that is a fully invariant subgroup of .

## Related facts

- Characteristicity is transitive
- Injective endomorphism-invariance is transitive
- Strict characteristicity is not transitive
- Normality is not transitive
- Homomorph-containment is not transitive

### Related facts about full invariance

- Fully invariant of strictly characteristic implies strictly characteristic
- Full invariance is quotient-transitive
- Full invariance does not satisfy intermediate subgroup condition
- Full invariance is strongly intersection-closed
- Full invariance is strongly join-closed

## Facts used

## 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:

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