Question

Prove that H c G is a normal subgroup if and only if every left coset is a right coset, i.e., aH = Ha for all a e G

Answer 1

`\Rightarrow`

Suppose first that
`H\subset G` is a normal subgroup. Then by definition we must have for all
`a\in H`,
`xax^(-1) \in H` for every
`x\in G`. Let
`a\in G` and choose
`(ab)\in aH` (
`b\in H`). By hypothesis we have
`aba^(-1) =abbb^(-1)a^(-1)=(ab)b(ab)^(-1) \in H`, i.e.
`aba^(-1)=c` for some
`c\in H`, thus
`ab=ca \in Ha`. So we have
`aH\subset Ha`. You can prove
`Ha\subset aH` in the same way.

`\Leftarrow`

Suppose
`aH=Ha` for all
`a\in G`. Let
`h\in H`, we have to prove
`aha^(-1) \in H` for every
`a\in G`. So, let
`a\in G`. We have that
`ha^(-1) =a^(-1)h'` for some
`h'\in H` (by the hypothesis). hence we have
`aha^(-1)=h' \in H`. Because
`a` was chosen arbitrarily we have the desired .