Order of Subgroup Product/Lemma
Jump to navigation
Jump to search
Lemma for Order of Subgroup Product
Let $h_1, h_2 \in H$.
Then:
- $h_1 K = h_2 K$
- $h_1$ and $h_2$ are in the same left coset of $H \cap K$.
Proof
Let $h_1, h_2 \in H$.
Then:
\(\ds h_1 K\) | \(=\) | \(\ds h_2 K\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds h_1^{-1} h_2\) | \(\in\) | \(\ds K\) | Left Cosets are Equal iff Product with Inverse in Subgroup | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds h_1^{-1} h_2\) | \(\in\) | \(\ds H \cap K\) | Definition of Set Intersection | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds h_1 \paren {H \cap K}\) | \(=\) | \(\ds h_2 \paren {H \cap K}\) | Left Cosets are Equal iff Product with Inverse in Subgroup |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $10$