- #1

jackmell

- 1,807

- 54

## Homework Statement

Let ##G## be a group and ##\sim## and equivalence relation on ##G##. Prove that if ##\sim## respects multiplication, then ##\sim## is the equivalence relation associated to some normal subgroup ##N\trianglelefteq G##; i.e., prove there is a normal subgroup ##N## such that ##x\sim y## iff ##xN =yN##.

## Homework Equations

An equivalence relation respects multiplication if ##x\sim y## implies that ## xz\sim yz## for all ##z\in G##.

## The Attempt at a Solution

I can show that if the equivalence is a partition of ##G## into cosets ## gH## with ## H\leq G## (H a subgroup of G), then in order for that equivalence to respect multiplication, ##H## has to be normal in ##G##:

Define the following equivalence relation on ##G##:

##

x\sim y \Rightarrow x\in yH\Rightarrow xH=yH,\quad H\leq G

##

where ##yH## is the representative for this coset in the partitioning of ##G##. Then in order for this equivalence to respect multiplication, we would need ##xzH=yzH\;\forall\; z\in G## or ##\left(yz\right)^{-1} (xz) H\in H## . That means ##z^{-1}\left(y^{-1}x\right)z\in H\;\forall\;z\in G##. Now the quantity ##y^{-1}x## represents all of ##H## so that is equivalent to ##z^{-1} H z\in H## and that would mean ##H## has to be normal.

However I don't understand how I can prove that every type of partitioning of ##G## into equivalent classes that respect multiplication necessarily is associated with some normal group or perhaps can be related to this coset partitioning.

Ok, thanks for reading,

Jack