Automorphism group

An automorphism of G is an isomorphism G → G. Under composition these form the automorphism group Aut(G).

Inner, outer

Each g ∈ G conjugates: x ↦ g x g⁻¹. These are the inner automorphisms,

Inn(G) ≅ G / Z(G),

the quotient by the center Z(G) (the kernel of g ↦ conjugation is exactly Z(G)). Inn(G) is a normal subgroup of Aut(G), and the outer automorphism group is the quotient

Out(G) = Aut(G) / Inn(G).

So Aut(G) is an extension of Out(G) by Inn(G),

1 → Inn(G) → Aut(G) → Out(G) → 1,

with |Inn(G)| = |G| / |Z(G)| and |Aut(G)| = |Inn(G)| · |Out(G)|. This extension need not split, so in general Aut(G) is not a semidirect product Inn(G) ⋊ Out(G).

Key facts

• G abelian ⟹ Z(G) = G ⟹ Inn(G) = {e} ⟹ Out(G) = Aut(G).
• Z(G) = {e} (centerless) ⟹ Inn(G) ≅ G.
• G is complete ⟺ Z(G) = {e} and Out(G) = {e} ⟺ G ≅ Aut(G) inner — e.g. every Sₙ, n ≠ 2, 6.

Examples

See also

• Center — Inn(G) = G/Z(G).
• Quotient group, Extension — how Aut sits over Out.
• Wikipedia: Automorphism group
• Wikipedia: Outer automorphism group