Cosets are Equivalent/Proof 2

Theorem

All left cosets of a group $G$ with respect to a subgroup $H$ are equivalent.

That is, any two left cosets are in one-to-one correspondence.

The same applies to right cosets.

As a special case of this:

$\forall x \in G: \order {x H} = \order H = \order {H x}$

where $H$ is a subgroup of $G$.

Proof

Follows directly from Set Equivalence of Regular Representations.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem