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

$$gS{g}^{-1}=\{gs{g}^{-1}\mathrm{⋮}s\in S\}.$$

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

$⟨S^g\mathrm{⋮}g\in G⟩:=S^G$(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