Congruence Modulo Subgroup is Equivalence Relation

Theorem

Let $G$ be a group, and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let $x \equiv^l y \pmod H$ denote the relation that $x$ is left congruent modulo $H$ to $y$.

Then the relation $\equiv^l$ is an equivalence relation.

Let $x \equiv^r y \pmod H$ denote the relation that $x$ is right congruent modulo $H$ to $y$

Then the relation $\equiv^r$ is an equivalence relation.

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 20$. Cosets: Theorem $33$