Conjugation (group theory): Difference between revisions

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(relate to commutator, conjugacy and conjugacy class)
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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:


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and so measures the failure of ''x'' and ''y'' to commute.
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 classess]]''.
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.