Conjugation (group theory): Difference between revisions

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


:<math>x^y = y^{-1} x y . \,</math>
:<math>x^y = y^{-1} x y . \,</math>
If ''x'' and ''y'' [[commutativity|commute]] then the conjugate of ''x'' by ''y'' is just ''x'' again.  The [[commutator]] of ''x'' and ''y'' can be written as
:<math>[x,y] = x^{-1} x^y , \, </math>
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 (mathematics)|relation]] of ''[[conjugacy]]'' is an [[equivalence relation]], whose [[equivalence class]]es are the ''[[conjugacy class]]es''.
==Inner automorphism==
For a given element ''y'' in ''G'' let <math>T_y</math> denote the operation of conjugation by ''y''.
It is easy to see that the [[function composition]] <math>T_y \circ T_z</math> is just <math>T_{yz}</math>.
Conjugation <math>T_y</math> preserves the group operations:
:<math>T_y(1) = 1^y = y^{-1} 1 y = 1 ; \,</math>
:<math>T_y(uv) = y^{-1}uvy = y^{-1}uyy^{-1}vy = u^y v^y = T_y(u) T_y(v) ; \,</math>
:<math>T_y(u)^{-1} = (y^{-1} u y)^{-1} = y^{-1}u^{-1}y = T_y(u)^{-1} . \, </math>
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.