Question

If is subgroup of G , prove that $\mathit{A}\mathbf{⊑}\mathit{N}\left(A\right)$.

Short answer

It is proved that$A⊑N\left(A\right)$

Step-by-step solution

Subgroup of a group

Let G be a group and H be a subgroup of a group G if for every $\mathit{a}\mathbf{,}\mathit{b}\in \mathit{H}\Rightarrow \mathit{a}\mathit{b}\in \mathit{H}\mathbf{}\mathbf{\&}\mathbf{}{\mathit{a}}^{-1}\in \mathit{H}$.

Normalizer of a group

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

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

Therefore, $N\left(A\right)$ consist all those elements of G such that $g^{-1}Ag=A$

And A is subgroup of G

Therefore, $A⊑N\left(A\right)$.

Hence proved.