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Suppose H is a subgroup of G. For g in G, define f_{g}: G/H > G/H by f_{g}(aH) = gaH for a in G, where G/H is the set of left cosets of H in G.

What is the difference between these two statements:

1) for a given aH in G/H, the set {g in G : f_{g}(aH) = aH }

2) set {g in G : f_{g}= the identity permutation in G/H}

The identity permutation, in this case, meaning f_{g}(aH) = gaH = aH for all cosets aH

I know that in part 1, a is given and so we can use a to find the solution set of g, but I struggle to work with part 2 without any concrete information about such an a.

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# Cosets: difference between these two statements

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