Equivalent Statements for Congruence Modulo a Subgroup

From ProofWiki

Jump to: navigation, search

Theorem

Let $G$ be a group, and let $H$ be a subgroup of $G$.

Let $x \ \equiv^l \ y \ \left({\bmod H}\right)$ denote that $x$ is left congruent modulo $H$ to $y$.

Then the following statements are equivalent:

$(1):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x$	$\equiv^l$	$\displaystyle y \ \left({\bmod H}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(2):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x^{-1} y$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(3):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \exists h \in H: x^{-1} y$	$=$	$\displaystyle h$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(4):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \exists h \in H: y$	$=$	$\displaystyle x h$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Let $G$ be a group, and let $H$ be a subgroup of $G$.

Let $x \ \equiv^r \ y \ \left({\bmod H}\right)$ denote that $x$ is right congruent modulo $H$ to $y$.

Then the following statements are equivalent:

$(1):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x$	$\equiv^r$	$\displaystyle y \ \left({\bmod H}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(2):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x y^{-1}$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(3):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \exists h \in H: x y^{-1}$	$=$	$\displaystyle h$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(4):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \exists h \in H: x$	$=$	$\displaystyle h y$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Each statement follows directly from the previous one, by definition of Congruence Modulo a Subgroup.

$\blacksquare$

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

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x\)	\(\equiv^r\)	\(\displaystyle y \ \left({\bmod H}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x y^{-1}\)	\(\in\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((3):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \exists h \in H: x y^{-1}\)	\(=\)	\(\displaystyle h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((4):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \exists h \in H: x\)	\(=\)	\(\displaystyle h y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)