Normal subgroup question!! Please help!!

Let G be a group and let H and K be subgroups of G with H is a normal subgroup of G. Define the subset of G

HK = {hk : h in H and k in K}

(a) Show that HK is a subgroup of G

(b) If K is also normal in G, show that HK is a normal subgroup of G

(a) first we need to show that HK is closed under multiplication. that is:

for h,h' in H, and k,k' in K, we need to show that:

(hk)(h'k') is in HK.

now since H is normal, for any k in K, kH = Hk.

in particular, kh' = h"k, for some h" in H, so:

(hk)(h'k') = h(kh')k' = h(h"k)k' = (hh")(kk'), which is in HK.

secondly, we must show that if hk is in HK, then so is:

(hk)^-1 = k^-1h^-1.

now k^-1H = Hk^-1 (since H is normal), so k^-1h^-1 = h'k^-1,

for some h' in H, and h'k^-1 is in HK.

(b) suppose H,K are normal.

then g(hk)g^-1 = (ghg^-1)(gkg^-1)

= h'k', for some h' in H (since ghg^-1 = h', since H is normal)

and k' in K (since gkg^-1 = k', since K is normal),

which shows that gHKg^-1 is contained in HK, so HK is normal.