# Normal-extensible not implies inner

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal-extensible automorphism) neednotsatisfy the second subgroup property (i.e., inner automorphism)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about normal-extensible automorphism|Get more facts about inner automorphism

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal-extensible automorphism but not inner automorphism|View examples of subgroups satisfying property normal-extensible automorphism and inner automorphism

## Statement

We can have a group and a normal-extensible automorphism of that is not an inner automorphism.

In fact, there exist groups for which it is *not* true that every automorphism is inner, but for which it is still true that every automorphism is normal-extensible.

## Facts used

## Proof

Given fact (1), we simply need to exhibit a group satisfying the condition of being centerless and maximal in its automorphism group. One example is the alternating group of degree four.