Complete group
A group G is complete if it is centerless and every automorphism is inner:
- Z(G) = {e}, and
- Out(G) = Aut(G)/Inn(G) = {e}.
Equivalently, the natural map G → Aut(G) sending g to conjugation by g is an isomorphism (injective ⟺ Z(G) = 1; surjective ⟺ Out(G) = 1). So a complete group is canonically isomorphic to its own automorphism group.
Key facts
- The two conditions are independent: C₂ has Out = 1 but Z = C₂ ≠ 1;
A₅has Z = 1 but Out = C₂ — neither is complete. - G ≅ Aut(G) by itself does not imply complete:
D₄satisfies Aut(D₄) ≅ D₄, yet Z(D₄) = C₂, so D₄ is not complete. - If a complete group K is normal in H, then it is a direct factor: H = K × C_H(K).
- G is complete ⟺ its holomorph splits as a direct square: Hol(G) = G ⋊ Aut(G) ≅ G × G.
- For a nonabelian simple group S, Aut(S) is complete.
Examples
S₃, order 6 — the smallest complete group: Z(S₃) = {e} and Aut(S₃) ≅ S₃.S₄(order 24) andS₅(order 120).- The symmetric group Sₙ is complete for every n ≥ 3 except n = 6.
Non-examples
- Every nontrivial abelian group: Z(G) = G ≠ {e}.
S₆: the exceptional symmetric group, with an outer automorphism (Out(S₆) = C₂).A₅and other nonabelian simple groups: centerless but with Out ≠ {e}.