Conjugate elements
in a group
Elements and of for which
for some in . One also says that is the result of conjugating by . The power notation is frequently used for the conjugate of under .
Let be two subsets of a group , then denotes the set
For some fixed in and some subset of the set is said to be conjugate to the set in . In particular, two subgroups and are called conjugate subgroups if for some in . If a subgroup coincides with for every (that is, consists of all conjugates of all its elements), then is called a normal subgroup of (or an invariant subgroup, or, rarely, a self-conjugate subgroup).
Comments
The map $x \mapsto g^{-1} x g$ for given $g$ is conjugation by $g$: it is an inter automorphism of $G$.
References
[a1] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) |
[a2] | D. Gorenstein, "Finite groups" , Chelsea, reprint (1980) |
Conjugate elements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_elements&oldid=35117