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}.

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