Equivalent Statements for Congruence Modulo Subgroup/Left
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Theorem
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $x \equiv^l y \pmod H$ denote that $x$ is left congruent modulo $H$ to $y$.
Then the following statements are equivalent:
\(\text {(1)}: \quad\) | \(\ds x\) | \(\equiv^l\) | \(\ds y \pmod H\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds x^{-1} y\) | \(\in\) | \(\ds H\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \exists h \in H: \, \) | \(\ds x^{-1} y\) | \(=\) | \(\ds h\) | ||||||||||
\(\text {(4)}: \quad\) | \(\ds \exists h \in H: \, \) | \(\ds y\) | \(=\) | \(\ds x h\) |
Proof
\(\ds x\) | \(\equiv^l\) | \(\ds y \pmod H\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x^{-1} y\) | \(\in\) | \(\ds H\) | Definition of Left Congruence Modulo Subgroup | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists h \in H: \, \) | \(\ds x^{-1} y\) | \(=\) | \(\ds h\) | Definition of Element of $H$ | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists h \in H: \, \) | \(\ds y\) | \(=\) | \(\ds x h\) | Division Laws for Groups |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.2$ Another approach to cosets