# Transitive normality is not centralizer-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., transitively normal subgroup)notsatisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property).

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## Contents

## Statement

### Statement with symbols

Suppose is a transitively normal subgroup of a group . Then, it is not necessary that (the centralizer of in ) is transitively normal.

## Related facts

### Related centralizer-closed properties

- Normality is centralizer-closed: Since transitively normal implies normal, this shows that the centralizer of a transitively normal subgroup, even though not necessarily transitively normal, is still normal.
- Central factor is centralizer-closed: Since central factor implies transitively normal, this shows that the centralizer of a central factor is transitively normal.

### Related metaproperty dissatisfactions for transitively normal subgroups

- Transitive normality is not finite-intersection-closed
- Transitive normality is not finite-join-closed

## Proof

`Further information: symmetric group:S3`

Suppose is a cyclic group of order three, is a symmetric group of degree three, and is the subgroup of order three in . Define:

.

Then, is a normal subgroup of prime order, hence is transitively normal. On the other hand, the centralizer of in is , which is not transitively normal. To see this, let be an isomorphism from to . Consider:

.

is a normal subgroup of (since is abelian). On the other hand, conjugating an element of by an element of sends to , which is not in . Hence, is not normal in . Thus, is not transitively normal in .