Conjugation (group theory): Difference between revisions
Jump to navigation
Jump to search
imported>Richard Pinch (relate to commutator, conjugacy and conjugacy class) |
imported>Richard Pinch m (typo) |
||
| Line 9: | Line 9: | ||
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 | 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''. | ||
Revision as of 12:59, 15 November 2008
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
![{\displaystyle [x,y]=x^{-1}x^{y},\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c04fa1883bb53d3a0894d9dcbd4978ad24611a0)
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.