Element in Left Coset iff Product with Inverse in Subgroup
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Theorem
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $x, y \in G$.
Let $y H$ denote the left coset of $H$ by $y$.
Then:
- $x \in y H \iff x^{-1} y \in H$
Proof
\(\ds x\) | \(\in\) | \(\ds y H\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists h \in H: \, \) | \(\ds x\) | \(=\) | \(\ds y h\) | Definition of Left Coset | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists h \in H: \, \) | \(\ds x^{-1}\) | \(=\) | \(\ds h^{-1} y^{-1}\) | Inverse of Group Product | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists h \in H: \, \) | \(\ds x^{-1} y\) | \(=\) | \(\ds h^{-1}\) | Product with $y$ on the right | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x^{-1} y\) | \(\in\) | \(\ds H\) | $H$ is a subgroup |
$\blacksquare$
Also see
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.6 \ \text {(1L)}$ Another approach to cosets
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $7 \ \text{(i)}$