Cosets: difference between these two statements

[ZZ Specs]

Hi all,

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.

 

[R136a1]

In (2) you demand $f_g(aH) = aH$ for all $a$. So it is the intersection of sets in (1).

 

[ZZ Specs]

Oh of course. Wow, I didn't see that. So the solution set could be considered:

Assuming a_i in G for all i in the index set I,
{g in G : g lies in the intersection ∏_{i in I} {g = a_i h a_i^-1 for some h in H} }

Not sure if ∏ is the best symbol to represent intersection, but for now let's go with it.

Thanks for the reply!