GAP — make an Additive Group into a Group

Question

I have a list of vectors $\texttt{A}$ with coordinates in $\{0,1,\ldots, n-1\}$ that happen to make a group under addition. How do I make GAP understand that it is a group?

One possible solution that my friend found is to use $\texttt{B := A*One(Integers mod n)}$ and then $\texttt{C := AdditiveGroup(B)}$. This gives an additive group (which for GAP is just an additive magma-with-zero and inverses) and I can compute sums of elts etc., but I can't use the standard functions for groups (such as $\texttt{StructureDescription(C)}$). Indeed, when you ask $\texttt{IsGroup(C)}$ the answer is $\texttt{false}$.

I feel like there should be a good way to make GAP treat that as a group, I checked $\texttt{AsGroup(C)}$, but it didn't work. Any ideas how to do it?

Answer 1

GAP only supports multiplicative groups. To deal with this, you need to construct an isomorphic multiplicative groups. I'll describe below how to do that.

You group C is isomorphic to $C_n^d$, where $d$ is the length of your vectors. In GAP, there are are many ways to construct these groups, e.g. like so:

G := AbelianGroup(ListWithIdenticalEntries(d, n));

We could also have used DirectProduct plus CyclicGroup; or asked for the groups in different representations.

Now we need a way to map the $d$-tuples generating your group $C$ into the group $G$. We can do that as follows (were we use that in this particular case, the generators of $G$ set by GAP happen to correspond to the $d$ factors of $C_n^d$):

gens := GeneratorsOfGroup(G);
# define a function mapping vectors to group elements
f := v -> Product([1..d], i -> gens[i]^Int(v[i]));

Using $f$, we can map your set $A$ into a subset of $G$, and then study the group generated by that, which is isomorphic to $C$

Let's combine this to a concrete example:

n := 12;; d := 3;;
A := List([1..2], i->List([1..d],j->Random(0,n-1)));
G := AbelianGroup(ListWithIdenticalEntries(d, n));
gens := GeneratorsOfGroup(G);
f := v -> Product([1..d], i -> gens[i]^v[i]);
H := Subgroup(G, List(A, f));
Size(H);
StructureDescription(H);

If we run this, we might get something like this (depending on the state of the random number generator):

gap> n := 12;; d := 3;;
gap> A := List([1..2], i->List([1..d],j->Random(0,n-1)));
[ [ 4, 3, 9 ], [ 10, 9, 11 ] ]
gap> G := AbelianGroup(ListWithIdenticalEntries(d, n));
<pc group of size 1728 with 3 generators>
gap> gens := GeneratorsOfGroup(G);
[ f1, f4, f7 ]
gap> f := v -> Product([1..d], i -> gens[i]^v[i]);
function( v ) ... end
gap> H := Subgroup(G, List(A, f));
Group([ f3*f4*f5*f7*f9^2, f2*f3^2*f4*f6^2*f7*f8*f9^2 ])
gap> Size(H);
72
gap> StructureDescription(H);
"C12 x C6"