# properties of conjugacy

Let $S$ be a nonempty subset of a group $G$.  When $g$ is an element of $G$, a conjugate of $S$ is the subset

 $gSg^{-1}\;=\;\{gsg^{-1}\,\vdots\;\;s\in S\}.$

We denote here

 $\displaystyle gSg^{-1}\;:=\;S^{g}.$ (1)

If $T$ is another nonempty subset and $h$ another element of $G$, then it’s easily verified the formulae

• $(ST)^{g}\;=\;S^{g}T^{g}$

• $(S^{g})^{h}\;=\;S^{gh}$

The conjugates $H^{g}$ of a subgroup $H$ of $G$ are subgroups of $G$, since any mapping

 $x\mapsto gxg^{-1}$

is an automorphism (an inner automorphism) of $G$ and the homomorphic image of group is always a group.

The notation (1) can be extended to

 $\displaystyle\langle S^{g}\,\vdots\;\;g\in G\rangle\;:=\;S^{G}$ (2)

where the angle parentheses express a generated subgroup.  $S^{G}$ is the least normal subgroup of $G$ containing the subset $S$, and it is called the normal closure of $S$.

http://en.wikipedia.org/wiki/ConjugacyWiki

Title properties of conjugacy PropertiesOfConjugacy 2013-03-22 18:56:35 2013-03-22 18:56:35 pahio (2872) pahio (2872) 5 pahio (2872) Topic msc 20A05 NormalClosure2 NonIsomorphicGroupsOfGivenOrder normal closure