Left Coset Equals Subgroup iff Element in Subgroup

Theorem

Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

Let $x \in G$.

Let $x H$ denote the left coset of $H$ by $x$.

Then:

$x H = H \iff x \in H$

Proof

\(\ds x H\)

\(=\)

\(\ds H\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x H\)

\(=\)

\(\ds e H\)

Left Coset by Identity: $e H = H$

\(\ds \leadstoandfrom \ \ \)

\(\ds x e^{-1}\)

\(\in\)

\(\ds H\)

Left Cosets are Equal iff Product with Inverse in Subgroup

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds H\)

Group Properties

$\blacksquare$

Also see

• Right Coset Equals Subgroup iff Element in Subgroup

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Example $112$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.6 \ \text {(3L)}$ Another approach to cosets
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Exercise $2$