Right Coset Space forms Partition

Theorem

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

The right coset space of $H$ forms a partition of its group $G$:

\(\ds x \equiv^r y \pmod H\) \(\iff\) \(\ds H x = H y\)
\(\ds \neg \paren {x \equiv^r y} \pmod H\) \(\iff\) \(\ds H x \cap H y = \O\)


Proof

Follows directly from:

Right Congruence Modulo Subgroup is Equivalence Relation
Relation Partitions Set iff Equivalence.

$\blacksquare$


Also see


Sources

  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Theorem
  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.1$
  • 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.4$ Another approach to cosets
  • 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem
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