I was reading about cosets and, given the fact that if $H$ is a non empty subset of a finite group G, we have the following equality $[G:H]|H|=|G|$, I came up with the following question:
If the order of a finite group $G$ is $|G|=n$, where $n$ is a composite number, and given $a,b$ positive divisors of $n$ wih $n=ab$ and $1<a,b<n$, does it always exist an equivalence relation $R$ defined on $G$ such that each equivalence class has exactly $a$ elements and there are $b$ different equivalence classes?
You certainly can always construct such an equivalence relation by partitioning $G$ to $b$ disjoint subsets of size $a$. There is no guarantee that this relation will be a congruence however.