TITLE: Question about Isomorphism of groups
QUESTION [7 upvotes]: The similarity of the third isomorphism theorem ($G/H\simeq(G/K)/(H/K)$) to elementary manipulation of fractions leads me to question when it is the case that $NK/HK\simeq N/H$ for subgroups $N,H,K$ of some group $(G,\circ)$. Thank you in advance.

REPLY [4 votes]: I'm going to assume $H$ is a normal subgroup of $N$ (so the quotient makes sense) and that $N$ and $H$ normalize $K$ (so that the products $NK$ and $HK$ are subgroups and $HK$ is normal in $NK$). There is an obvious surjective map $N/H \rightarrow NK/HK$. Its kernel is $(HK \cap N)/H$. This explains why (and exactly when) examples such as DonAntonio's occur: the map is an isomorphism precisely when $HK \cap N=H$.

REPLY [3 votes]: No:
$$C_6=\langle x\rangle=\{1,c,c^2,...,c^5\}\;,\;\;H=\langle c^2\rangle\;,\;K=\langle c^3\rangle\;,\;N=G\implies$$
$$NK/HK=G/HK=G/G\cong\{1\}\;\;,\;\;N/H=G/H\cong C_2\ldots$$