Given $H$ and $K$ are normal subgroups in $G$ such that $H \bigcap K=${$1_G$}
SHOW: $xy = yx$ for all $x \in H$ and $y \in K$
This is what I have so far:
$x \in H$ and $y \in K$
$\therefore g_{1}xg_{1}^{-1}=x$
and $g_{2}yg_{2}^{-1}=y$
for $g_{1}, g_{2} \in G$
$\therefore xy=g_{1}xg_{1}^{-1}g_{2}yg_{2}^{-1}$
$=g_{2}yg_{2}^{-1}g_{1}xg_{1}^{-1}$
$=yx$
is this correct?