Inner automorphism group

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Definition

Symbol-free definition

The inner automorphism group of a group is defined in the following equivalent ways:

  • It is the subgroup of the automorphism group comprising those automorphisms that are inner, viz those automorphisms that arise as conjugation by an element.
  • It is the quotient of the group by its center.

Definition with symbols

The inner automorphism group of a group G, denoted as Inn(G), is defined in the following equivalent ways:

  • It is the subgroup of Aut(G) comprising those automorphisms that are inner, viz those automorphisms that arise as conjugation by an element. That is, it is the set:

\{\sigma \in Aut(G)|\exists g \in G , \sigma(x) = gxg^{-1} \forall x \in G \}

  • It is the quotient G/Z(G) where Z(G) denotes the center of G. In other words, it is the set of equivalence classes in G under the relation of their ratio being an element in the center.

Equivalence of definitions

A group acts on itself as automorphisms by conjugation. Thus, there is a natural homomorphism c: G \to Aut(G) that sends g to the automorphism c_g = x \mapsto gxg^{-1}. The kernel of c is the center of G.

The inner automorphism group is the image of this homomorphism. Note that when we view it as a subgroup of Aut(G), we get the first definition. When viewed as the quotient of G by the kernel, we get the second definition.

Facts

Realization as inner automorphism group

Every group may not be realized as the inner automorphism group of some group. A group H is termed a capable group if there is a group G such that H = G/Z(G).

Properties of the inner automorphism group

A group whose inner automorphism group is Abelian is termed a nilpotence class-2 group.

A group is nilpotent if and only if its inner automorphism group is nilpotent.