Question

If A is subgroup of G, prove that $g\in \left(A\right)$ if and only if $Ag=gA$ .

Short answer

It is proved that $g\in N\left(A\right)$ if and only if$Ag=gA$

Step-by-step solution

Normalizer of subgroup

Let A be a subgroup of a group G . The normalizer of A is the set $N\left(A\right)$ defined by

$N\left(A\right)=\left\{g\in G/g^{-1}Ag=A\right\}$

Now, suppose $g\in N\left(A\right)$

By, definition,

$\begin{array}{rcl}{g}^{-1}Ag& =& A\\ & \Rightarrow & g{g}^{-1}Ag=gA\\ Ag& =& gA\end{array}$

Proving converse part

Conversely, let $Ag=gA$, this implies that:

$$\begin{gathered} g^{-1}Ag=g^{-1}gA \\ {g}^{-1}Ag=A \\ g\in N\left(A\right) \end{gathered}$$

Hence proved.