## My first post – cosets

This is my first post to my wordpress blog.

Let ${G}$ be a group and H a subgroup of ${G}$. A left coset of ${H}$ generated by the element ${ x \in G }$ is the set constructed in this way: For each element ${x }$of G, we consider all the elements of the form ${xH}$, where ${ h }$ runs over all elements of the subgroup ${H}$.

Exercise: Suppose that an element ${y }$ belongs to the left coset of ${H}$ generated by an element ${x }$. Prove that the left cosets of ${H}$ generated by elements ${ x }$ and ${y}$ coincide.

Proof: We know that ${ y= xh_1 }$ ,where ${ h_1 }$ is an element of the subgroup ${H}$. Hence ${ x= y h_1^{-1}}$. Let ${ h }$ be an arbitrary element of the subgroup ${ H}$. The elements ${ h_1h}$ and ${ h_1^{-1} h }$ belong to ${H}$. Hence ${yh=(xh_1)h=x(h_1h) }$ belongs to ${ x H }$, and the element ${ xh=(yh_1^{-1})h =y(h_1^{-1}h) }$ belongs to ${ yh}$. Since every element of ${yH}$ belongs to ${ xh}$ and vice versa, it follows that ${ xH=yH}$.

$\Box$