Conjugate elements
in a group $G$
Elements and
of
for which
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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