Equivalent Statements for Congruence Modulo Subgroup

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $G$ be a group.

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


Left Congruence

Let $x \equiv^l y \pmod H$ denote that $x$ is left congruent modulo $H$ to $y$.

Then the following statements are equivalent:

\(\text {(1)}: \quad\) \(\ds x\) \(\equiv^l\) \(\ds y \pmod H\)
\(\text {(2)}: \quad\) \(\ds x^{-1} y\) \(\in\) \(\ds H\)
\(\text {(3)}: \quad\) \(\ds \exists h \in H: \, \) \(\ds x^{-1} y\) \(=\) \(\ds h\)
\(\text {(4)}: \quad\) \(\ds \exists h \in H: \, \) \(\ds y\) \(=\) \(\ds x h\)


Right Congruence

Let $x \equiv^r y \pmod H$ denote that $x$ is right congruent modulo $H$ to $y$.

Then the following statements are equivalent:

\(\text {(1)}: \quad\) \(\ds x\) \(\equiv^r\) \(\ds y \pmod H\)
\(\text {(2)}: \quad\) \(\ds x y^{-1}\) \(\in\) \(\ds H\)
\(\text {(3)}: \quad\) \(\ds \exists h \in H: \, \) \(\ds x y^{-1}\) \(=\) \(\ds h\)
\(\text {(4)}: \quad\) \(\ds \exists h \in H: \, \) \(\ds x\) \(=\) \(\ds h y\)