Group acts as automorphisms by conjugation

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This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article describes a pivotal group action: a group action on a set closely associated with the group. This action is important to understand and remember.
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Statement

Let be a group. For any , define the map:

given by:

(this is termed conjugation by or the inner automorphism induced by ).

Then, the following are true:

Related facts

Analogues in other structures

Proof

Proof that every conjugation is an automorphism

Given: A group , an element

To prove: The map is an automorphism of

Proof: is a map from to . We need to thus prove three things:

The key idea for this is the fact that . Formally:

The key idea for this is the fact that . Formally:

Here, we use the fact that the inverse of a product of elements is the product of their inverses, in reverse order (i.e. the inverse map is involutive). Thus, we have:

Note that for the proof to go through we need to use the fact that and are left and right inverses of each other.

NOTE: It actually suffices to prove only the first of these three things, because to test whether a map between groups is a homomorphism of groups, it suffices to check whether it sends products to products. However, when working in somewhat greater generality than groups, it becomes important to check the other conditions, and they're explained here for illustrative purposes.

Thus, every is a homomorphism. It remains to show that this homomorphism is injective and surjective. Injectivity is clear, because . For surjectivity, note that given any , setting:

yields . (This also becomes clearer from the fact that the map is itself a homomorphism of groups).

Thus, is an automorphism.

Proof that the map is a homomorphism

Given: A group

To prove: The map , is a homomorphism from to

Proof: We'll check three things:

The proof again uses the reversal law for inverses (inverse map is involutive):

  • is the identity map:

This follows from the first two facts, because which is the identity map, and the same holds for composition in the other order.