This is my first post to my wordpress blog.

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

Exercise: Suppose that an element belongs to the left coset of generated by an element . Prove that the left cosets of generated by elements and coincide.

Proof: We know that ,where is an element of the subgroup . Hence . Let be an arbitrary element of the subgroup . The elements and belong to . Hence belongs to , and the element belongs to . Since every element of belongs to and vice versa, it follows that .

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