Let $G$ be a group, $H$ is a subgroup of $G$; for $a,b \in G$ we say $a$ is congurent to $b \mod H$, written as $a \equiv b \mod H$ if $ab^{-1} \in H$
Herstein, Topics in abstract algebra, page 34.
What is the motivation behind the above definition?
My thoughts:
I am guessing it is doing something with the fact that a subgroup divides the group into a union of disjoint cosets of that subgroup. In modular arithmetic we say equivalence if remainders are same.. but I am having a hard time understanding in what sense the elements are thought of remainders here.
The usual notion of congruence in integers says that $a\equiv b \bmod n$ if and only $n\mid (a-b)$, or $a-b\in n\Bbb Z$.
If we take $G=(\Bbb Z,+)$ and $H=n\Bbb Z$, then $ab^{-1}=a-b$ and we can rewrite this as $$ a\equiv b \mod H \Longleftrightarrow ab^{-1}\in H. $$ This is a motivation to define this in general.
Actually, many concepts of group theory can be illustrated with the group of integers and this often has a nice interpretation from elementary number theory.