Inner automorphism group
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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 , denoted as , is defined in the following equivalent ways:
- It is the subgroup of comprising those automorphisms that are inner, viz those automorphisms that arise as conjugation by an element. That is, it is the set:
- It is the quotient where denotes the center of . In other words, it is the set of equivalence classes in 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 that sends to the automorphism . The kernel of is the center of .
The inner automorphism group is the image of this homomorphism. Note that when we view it as a subgroup of , we get the first definition. When viewed as the quotient of by the kernel, we get the second definition.
Realization as inner automorphism group
Every group may not be realized as the inner automorphism group of some group. A group is termed a capable group if there is a group such that .
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.