# Alg7.12

From Example Problems

Let be any subgroup of the group . The set of left cosets of in form a partition of . Furthermore, if and only if and in particular, if and only if and are representatives of the same coset.

because

Therefore so .

To show that distinct left cosets are disjoint, suppose and show that .

Let Then for some .

Multiply both sides of the second equation by

with

Now I can write as , so for some ,

This means .

Similarly and so . This finishes the first part of the proof.

To show if and only if ,

which means for some .

Since , if and only if and are representatives of the same coset.