Let G be a group
Let H and K be subgroups of G with H normal in G
define HK:= {xy|x in G ^ y in K}
prove H is a normal subgroup of HK

Question

Let G be a group
Let H and K be subgroups of G with H normal in G
define HK:= {xy|x in G ^ y in K}
prove H is a normal subgroup of HK

zzr0ck3r · Answer

proof: Let h be in H
then (xy)*h(xy)^-1 = xyhy^(-1)x^(-1) = xh'x^(-1) is in H 
this is because yhy^(-1) is in H because y is in G and xhx^(-1) is in H because x is in G

zzr0ck3r · Answer

this look ok?

nincompoop · Answer

the format doesn't fit the conventional notation of providing a proof :( 
there's a proof wiki, see if that'll help :)

nincompoop · Answer

found it http://www.proofwiki.org/wiki/Definition:Normal_Subgroup

zzr0ck3r · Answer

look at definition 3

zzr0ck3r · Answer

or better yet look at the also known as...

kinggeorge · Answer

Your proof looks fine to me. The one thing I have to add, is that you may need to show that $HK$ is a subgroup of $G$. In fact, this is true if $HK$ is a subset of the normalizer of $H$ in $G$. I.e., if $H\lhd G$, then $HK$ is a subgroup of $G$.