Hi there,

I'm going over some past papers for an exam but with no answers and have come a cropper on this one, any help appreciated!

Let $\displaystyle H$ be a subgroup of a group $\displaystyle G$ and let $\displaystyle g \in G$. Define

$\displaystyle H^g := \{ ghg^{-1} | h \in H \}$

i) Prove that $\displaystyle H^g$ is a subgroup of $\displaystyle G$.

ii) Let $\displaystyle g, k, \in G$. Which of these statements is true

$\displaystyle (H^g)^k = H^{gk} or (H^g)^k = H^{kg}$

Give a proof of the statement that is always true

iii) Set $\displaystyle N := \bigcap_{g \in G} H^g$. Prove that $\displaystyle N \triangleleft G$ and that $\displaystyle N$ is a subgroup of $\displaystyle G$ that is contained in $\displaystyle H$.

Pretty stuck except for ii) $\displaystyle (H^g)^k = H^{kg}$ is obviously the right one (I hope!).