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Abstract Algebra

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Equivalence relation on a group

ABSALG-RQE1VJ

Let $S$ be a set. A (binary) relation ${\mathcal R}$ on $S$ is a subset of the Cartesian product $S\times S$.

If $(a,b)\in{\mathcal R}$ we write $aRb$, and say that $a$ is related to $b$.

A relation ${\mathcal R}\subseteq S\times S$ is an equivalence relation if and only if, for all $s,t,u\in S$:

$sRs$ (reflexive)

$sRt$ implies $tRs$ (symmetric)

$sRt$ and $tRu$ implies $sRu$ (transitive)

Which of the following define equivalence relations on EVERY group $G$ with neutral element $e$, for every subgroup $H$ of $G$?

A

$(g,g')\in {\mathcal R}$ if and only if $\{gh\mid h\in H\}=\{g'h\mid h\in H\}$

B

$(g,g')\in {\mathcal R}$ if and only if $\{gh\mid h\in H\}=\{hg'\mid h\in H\}$

C

$(g,g')\in {\mathcal R}$ if and only if $gg'=g'g$

D

$(g,g')\in{\mathcal R}$ if and only if $(g(g')^{-1})^m=e$, for a fixed integer $m\not=0$