Difference between revisions of "Conjugate elements"
From Encyclopedia of Mathematics
(Comment: Conjugation by a given element) |
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| − | ''in a group | + | {{TEX|part}} |
| + | ''in a group $G$'' | ||
Elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250104.png" /> for which | Elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250104.png" /> for which | ||
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====Comments==== | ====Comments==== | ||
| − | The map $x \mapsto g^{-1} x g$ for given $g$ is ''conjugation by $g$'': it is an [[ | + | The map $x \mapsto g^{-1} x g$ for given $g$ is ''conjugation by $g$'': it is an [[inner 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 20:52, 29 November 2014
in a group $G$
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 inner 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=35115
Conjugate elements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_elements&oldid=35115
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article