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ZZ Specs
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Homework Statement
Suppose H is a subgroup of G. For g in G, define fg : G/H > G/H by fg (aH) = gaH for a in G, where G/H is the set of left cosets of H in G.
I know that fg is a well-defined permutation. However, we have not established (yet) that G/H is a group.
2 parts to the question:
1) for a given aH in G/H, find the set {g in G : fg(aH) = aH }
2) find the set {g in G : fg = the identity permutation in G/H}
The Attempt at a Solution
I have done part (1), finding the solution set {g in G : g = aha-1 for some h in H}.
However, I struggle with part (2), as we have no information on a or H so I'm not sure what counts as a solution. I feel that normality may be involved but I cannot find out how to use it.
I know we want g such that fg(aH) = gaH = aH for all cosets aH ; this is the identity permutation. By equality of cosets, we can say that a-1ga = h for some a in G and h in H, or that g = aha-1 for some a in H and g in G, but I'm not sure if this consitutes a solution.
Any help is very much appreciated. Thank you.