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
$$gS{g}^{1}=\{gs{g}^{1}\mathrm{\vdots}s\in S\}.$$ 
We denote here
$gS{g}^{1}:={S}^{g}.$  (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
$$x\mapsto gx{g}^{1}$$ 
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
$\u27e8{S}^{g}\mathrm{\vdots}g\in G\u27e9:={S}^{G}$  (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 

Canonical name  PropertiesOfConjugacy 
Date of creation  20130322 18:56:35 
Last modified on  20130322 18:56:35 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  5 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 20A05 
Related topic  NormalClosure2 
Related topic  NonIsomorphicGroupsOfGivenOrder 
Defines  normal closure 