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:

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


Proof

\(\displaystyle x\) \(\equiv^l\) \(\displaystyle y \pmod H\)
\(\displaystyle \iff \ \ \) \(\displaystyle x^{-1} y\) \(\in\) \(\displaystyle H\) Definition of Left Congruence Modulo Subgroup
\(\displaystyle \iff \ \ \) \(\displaystyle \exists h \in H: x^{-1} y\) \(=\) \(\displaystyle h\) Definition of Element of $H$
\(\displaystyle \iff \ \ \) \(\displaystyle \exists h \in H: y\) \(=\) \(\displaystyle x h\) Division Laws for Groups

$\blacksquare$


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