Congruence Modulo Subgroup is Equivalence Relation
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Theorem
Let $G$ be a group, and let $H$ be a subgroup of $G$.
Let $x, y \in G$.
Left Congruence Modulo Subgroup is Equivalence Relation
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.
Right Congruence Modulo Subgroup is 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.
Also see
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 20$. Cosets: Theorem $33$