# 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:

\(\text {(1)}: \quad\) | \(\displaystyle x\) | \(\equiv^l\) | \(\displaystyle y \pmod H\) | ||||||||||

\(\text {(2)}: \quad\) | \(\displaystyle x^{-1} y\) | \(\in\) | \(\displaystyle H\) | ||||||||||

\(\text {(3)}: \quad\) | \(\displaystyle \exists h \in H: x^{-1} y\) | \(=\) | \(\displaystyle h\) | ||||||||||

\(\text {(4)}: \quad\) | \(\displaystyle \exists h \in H: y\) | \(=\) | \(\displaystyle x h\) |

## 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 |

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 42.2$ Another approach to cosets