Left Coset Space forms Partition

Theorem

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

The left coset space of $H$ forms a partition of its group $G$, and hence:

\(\ds x \equiv^l y \pmod H\)

\(\iff\)

\(\ds x H = y H\)

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

\(\iff\)

\(\ds x H \cap y H = \O\)

Proof

Follows directly from:

Left Congruence Modulo Subgroup is Equivalence Relation

Relation Partitions Set iff Equivalence

$\blacksquare$

Also see

• Right Coset Space forms Partition

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 $11.1$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 37$
• 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: Corollary $5.5$