Can the Quotient Group $HK/K$ be Confusing? Let $H, K$ be subgroups of $G$.  If $K$ is normal to G, it can be shown that:


*

*$HK$ is a subgroup of G.

*$K$ is a normal subgroup of $HK$.


Therefore, $HK/K$ becomes a group with coset multiplication.  Now, since $H$ is arbitrary, $K$ may not be a normal subgroup of $H$.  So, $H/K$ may not even be a quotient group.  Yet every element in $HK/K$ looks like something from $H/K$:  $hkK = hK$.  Isn't that going to be a problem?
This question came up while I was learning about group isomorphism theorems.
 A: I asked myself a similar question once, maybe you like to take a look.
Maybe some remarks of myself over the answers given there. On the level of sets by defining $H / K := \{ hK : h \in K \}$ just as a set, both sets $H/K$ and $HK / K$ are indeed the same, but you should not think of theme just as sets, but as something with more structure. This is the semantic view of factor groups, and this is answered in the post of mine linked above.
To give more context, the quotient/factor group notation is so choosen as to convey such semantical meaning. To be more specific, a factor group $G / N$ is the same as a surjective homomorphism (take the projection, and vice verse, each surjective homomorphism induces a factor group by taking as $N$ the kernel).
And so if you have a subgroup $H$, the subgroup $HK$ (if $K \unlhd G$) is a subgroup generated such that $K \le HK$, and it is the inverse image of the projection mapping $\pi : G \to G/K$, i.e. $HK = \pi^{-1}(\pi(H))$. So writting $HK$ makes sense as
i) it gives you a group such that $K$ is contained in, i.e. it could be modded out,
ii) it gives you a group in the factor group for which the "correspondence principle" applies, i.e. subgroup in the factor groups correspond precisely to the subgroup in $G$ which contain $K$,
iii) it allows rearrangements like
$$
 UK / K = VK / K
$$
then $UK = VK$ (which, if you think about factor groups as surjective homomorphism, that if the images are the same, the preimages must equal too).
So what you write at the top of a quotient are the pre-images of the natural projections maps, so that all these correspondences and rearrangements make sense (and by the way they then just reduce to statements about homomorphisms).
