Right Cosets are Equal iff Element in Other Right Coset
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Theorem
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $x, y \in G$.
Let $H x$ denote the right coset of $H$ by $x$.
Then:
- $H x = H y \iff x \in H y$
Proof
\(\ds H x\) | \(=\) | \(\ds H y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x y^{-1}\) | \(\in\) | \(\ds H\) | Right Cosets are Equal iff Product with Inverse in Subgroup | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds H y\) | Element in Right Coset iff Product with Inverse in Subgroup |
$\blacksquare$
Also see
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Exercise $3$