Difference between revisions of "Conjugate elements"
(LaTeX part) |
(Comment: Conjugacy is an equivalence relation) |
||
Line 17: | Line 17: | ||
====Comments==== | ====Comments==== | ||
+ | Conjugacy of elements is an [[equivalence relation]] on $G$, and the equivalence classes are the ''conjugacy classes'' of $G$. | ||
+ | |||
The map $x \mapsto g^{-1} x g$ for given $g$ is ''conjugation by $g$'': it is an [[inner automorphism]] of $G$. | 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 20:53, 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
Conjugacy of elements is an equivalence relation on $G$, and the equivalence classes are the conjugacy classes of $G$.
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) |
Conjugate elements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_elements&oldid=35117