Conjugation (group theory)

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In group theory, conjugation is an operation between group elements. The conjugate of x by y is:

xy=y−1xy.

If x and y commute then the conjugate of x by y is just x again. The commutator of x and y can be written as

[x,y]=x−1xy,

and so measures the failure of x and y to commute.

Two elements are said to be conjugate if one is obtained as a conjugate of the other: the resulting relation of conjugacy is an equivalence relation, whose equivalence classes are the conjugacy classes.

## Inner automorphism[edit]

For a given element y in G let Ty denote the operation of conjugation by y. It is easy to see that the function composition Ty∘Tz is just Tyz.

Conjugation Ty preserves the group operations:

Ty(1)=1y=y−11y=1;
Ty(uv)=y−1uvy=y−1uyy−1vy=uyvy=Ty(u)Ty(v);
Ty(u)−1=(y−1uy)−1=y−1u−1y=Ty(u)−1.

Since Ty is thus a bijective function, with inverse function Ty−1, it is an automorphism of G, termed an inner automorphism. The inner automorphisms of G form a group Inn(G) and the map y↦Ty is a homomorphism from G onto Inn(G). The kernel of this map is the centre of G.