Let be a group, a subgroup, and a fixed element. Prove that if for some element , then and .

Suppose for some . In particular, ; then for some , so that . Left multiplying by we have , but since , , and we have . Moreover, so that .

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## Comments

Why is k an element of Hg? I agree that if kH=gH, then k is an element of gH (or for right cosets), but why is this true for “mixing” left and right cosets?

What is a coset?

is the set of all products of the form where . Now let .