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

Title	properties of conjugacy
Canonical name	PropertiesOfConjugacy
Date of creation	2013-03-22 18:56:35
Last modified on	2013-03-22 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