Namespaces
Variants
Actions

Difference between revisions of "Conjugate elements"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Comment: Conjugation by a given element)
Line 16: Line 16:
  
 
====Comments====
 
====Comments====
 +
The map $x \mapsto g^{-1} x g$ for given $g$ is ''conjugation by $g$'': it is an [[inter automorphism]] of $G$.
  
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Chelsea, reprint  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Chelsea, reprint  (1980)</TD></TR></table>

Revision as of 20:51, 29 November 2014

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)
How to Cite This Entry:
Conjugate elements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_elements&oldid=12462
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article