It would be great if someone could give me an example of a group such that the following happens
a $\equiv$ b (mod N) and c $\equiv$ d (mod N) but ac $\not\equiv$ bd (mod N)
where N is a subgroup of a group G and a $\in$ G and b $\in$ G
a $\equiv$ b (mod N) iff $ab^{-1}$ $\in N$.
I saw an example of in the $S_n$, however I want to see one more concrete example to help me form better intuition.
Take $G:=\mathbb{D}_4$ it is generated by a rotation $R$ and an axial symmetry $s$.
You know that $R^4=1$ and $sRs^{-1}=R^3$ and $s^2=1$.
Now we take $N:=<s>$. We set :
$$a:=c:=R \text{ and }b:=d:=Rs$$
Then :
$$a=c \text{ mod } N$$
$$b=d \text{ mod } N$$
But, on the other hand : $$ac=R^2 \text{ and }bd= RsRs= RR^3=R^4=1$$
Of course $ac\notin N$.