Find all homomorphisms from a Finite Field (using GAP)

Question

I'm currently working with $\mathbb F_{243}$, viewing it as an additive group. I'd like to have GAP calculate all of the (additive) homomorphisms from this group to smaller groups.

I've done this with other (multiplicative) groups as follows:

Ns := NormalSubgroups(G);
    
for N in Ns do 
     hom := NaturalHomomorphismByNormalSubgroup(G, N);
od;

This approach doesn't work for the above finite field though, as GAP views groups multiplicatively only. I guess I need to convert F := FiniteField(243) into an additive group G that GAP can understand, but I'm quite unsure as to how to do that (I'm reasonably new to GAP). Could anyone offer any assistance? Many thanks.

Answer 1

Okay I'm going to answer my own question because I've managed to cobble together something that works - in case anyone else finds it useful.

I'd still be quite happy to see if anyone has a better way, or sees an issue with this method though!

F := FiniteField(3, 5);
G := AbelianGroup(ListWithIdenticalEntries(5, 3));
basis := Basis(F);
gens := GeneratorsOfGroup(G);
f := v -> Product([1..5], i -> gens[i]^IntFFE(Coefficients(basis, v)[i]));

The finite field order 243 is additively isomorphic to $\mathbb Z_3^5$ (see: What do the elements of $\mathbb{F}_{243}$ look like?) so if we create that group and then map each element as a product of basis vectors to the corresponding product of generators we should get the additive isomorphism (the function f) we're looking for. I can then apply the homomorphism by normal subgroups method as above (there's a very fun 2664 of them...)