# Using generators to check for a normal subgroup

1. Mar 31, 2013

### gauss mouse

1. The problem statement, all variables and given/known data
Perhaps I should say first that this question stems from an attempt to show that in the group
$\langle x,y|x^7=y^3=1,yxy^{-1}=x^2 \rangle,\ \langle x \rangle$ is a normal subgroup.

Let $G$ be a group with a subgroup $H$. Let $G$ be generated by $A\subseteq G$. Suppose that $H$ is closed under conjugation by elements from $A$ in the sense that $aha^{-1}\in H$ for any $a\in A$ and any $h\in H$. Is it then true that $H$ is a normal subgroup of $G$?

3. The attempt at a solution
I know that $G$ is the set of all products of elements from $A\cup A^{-1}$ where $A^{-1}:=\{a^{-1}|a\in A\}.$ Since we are assuming that $aha^{-1} \in H$ for any $a\in A$ and any $h\in H$, it will be enough to show that $a^{-1}ha\in H$ for any $a\in A$ and any $h\in H$. However I have not been able to show this.