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.

Proof

\(\ds \)

\(\)

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

\(\ds \)

\(\leadsto\)

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

Definition of Congruence Modulo Subgroup

\(\ds \)

\(\leadsto\)

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

Congruence Modulo Subgroup is Equivalence Relation, therefore Symmetric

$\blacksquare$

Sources

• 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