Cosets in Abelian Group

Theorem

Let $G$ be an abelian group.

Then every right coset modulo $H$ is a left coset modulo $H$.

That is:

$\forall x \in G: x H = H x$

In an abelian group, therefore, we can talk about congruence modulo $H$ and not worry about whether it is left or right.

$\ds $

$\ds \forall x, y \in G: x^{-1} y = y x^{-1}$

$\ds $

$\leadsto$

$\ds \paren {x \equiv^l y \pmod H \iff y \equiv^r x \pmod H}$

Definition of Congruence Modulo Subgroup

$\ds $

$\leadsto$

$\ds \paren {x \equiv^l y \pmod H \iff x \equiv^r y \pmod H}$

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Example $113$
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 37 \gamma$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42$. Another approach to cosets