---
tag: 09HEXDQ0CQ
kind: definition
name: Outer automorphism group
msc: [20F28]
defines: 0Q8Y862DTZ
uses: [0CWXCWP0Y0, 0H47DMFN3S, 0YYWCNTPWR]
sources:
  - url: https://en.wikipedia.org/wiki/Outer_automorphism_group
    locator: lead section
---

## Definition

Let $G$ be a [[group|G5S1M9]].
The outer automorphism group of $G$ is the [[quotient group|0KYX2N69J0]] $\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G)$.

## Remarks

The quotient exists because $\operatorname{Inn}(G)$ is a normal subgroup of $\operatorname{Aut}(G)$.
An automorphism that is not inner is called an outer automorphism, but the elements of $\operatorname{Out}(G)$ are cosets, not automorphisms.
For an abelian group, $\operatorname{Inn}(G)$ is trivial, so $\operatorname{Out}(G) \cong \operatorname{Aut}(G)$.
The symmetric group $S_6$ is the only symmetric group $S_n$ with nontrivial outer automorphism group.

## Explanations

### L1 human

Some symmetries of a group come from conjugation by its own elements, and some do not.
The outer automorphism group keeps only the second kind, up to the first kind.
For many groups it is small or trivial.