Conjugation (group theory): Difference between revisions

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Since <math>T_y</math> is thus a [[bijective function]], with [[inverse function]] <math>T_{y^{-1}}</math>, it is  an [[automorphism]] of ''G'', termed an '''inner automorphism'''.  The inner automorphisms of ''G'' form a group <math>Inn(G)</math> and the map <math>y \mapsto T_y</math> is a homomorphism from ''G'' [[surjective function|onto]] <math>Inn(G)</math>.  The [[kernel of a homomorphism|kernel]] of this map is the [[centre of a group|centre]] of ''G''.
Since <math>T_y</math> is thus a [[bijective function]], with [[inverse function]] <math>T_{y^{-1}}</math>, it is  an [[automorphism]] of ''G'', termed an '''inner automorphism'''.  The inner automorphisms of ''G'' form a group <math>Inn(G)</math> and the map <math>y \mapsto T_y</math> is a homomorphism from ''G'' [[surjective function|onto]] <math>Inn(G)</math>.  The [[kernel of a homomorphism|kernel]] of this map is the [[centre of a group|centre]] of ''G''.[[Category:Suggestion Bot Tag]]

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

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

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

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

Conjugation preserves the group operations:


Since is thus a bijective function, with inverse function , it is an automorphism of G, termed an inner automorphism. The inner automorphisms of G form a group and the map is a homomorphism from G onto . The kernel of this map is the centre of G.