Why This Subset is not a Subgroup. So there's a group $G$ where it has subgroups $H$ and $K$. So $H\cap K =$ {$h*k : h\in H, k\in K$} is not a subgroup though.
For some reason I can't see why this isn't a subgroup at all. For a subset to be a subgroup, it has to be non-empty, be communicative and have an inverse... But it seems that all three pass. I can't see why, can anyone give me some sort of insight to why this isn't a subgroup?
 A: Perhaps you will find this example instructive. 
Let $G=S_3$, the group of permutations of 3 symbols. Let $H=\{\,1,(12)\,\}$ and let $K=\{\,1,(13)\,\}$. These are both subgroups of $G$ (as you can check). Now $$HK=\{\,(1)(1),(12), (13), (12)(13)\,\}=\{\,1,(12),(13),(123)\,\}$$ is not a subgroup --- for example, it contains $(123)$ but not the inverse of $(123)$ (which is $(132)$). 
A: $H\cap K$ is the intersection of $H$ and $K$, defined as $\{g\mid g\in H, g\in K\}$.  This is the not equivalent to the definition you give.  The definition you give is of the product of $H$ and $K$, written $HK$, as mentioned in the comments.  You also have the wrong definition of a subgroup.  The corrected definition of a subgroup of a group $G$ is simply a subset of $G$ that is itself a group.  Moreover, just as there are noncommutative (I assume that's what you meant to say instead of noncommunicative) groups, there are noncommutative subgroups.
So this may not be an answer but it's as close to an answer as there can be, considering this was not a properly formulated question.
