/***
*
* BracesWithAbelianGroup
*
* This function returns the list of left braces with finite additive
* group isomorphic to <A>.
*
***/
BracesWithAbelianGroup := function(A)
local used, G, f, g, H, p;
used := [];
p := [];
if not IsAbelian(A) then
return p;
end if;
G, f, g := Holomorph(A);
for c in Subgroups(G:OrderEqual:=Order(A), IsSolvable:=true, IsRegular:=true) do
for H in Conjugates(G, c`subgroup) do
if forall{ x : x in used | IsConjugate(Stabilizer(G, 1), H, x) eq false} then
Append(~used, H);
Append(~p, [ [a,h]: a in Image(f), h in H | a*h in Stabilizer(G, 1) ]);
end if;
end for;
end for;
return p;
end function;
/***
*
* AllBraces
*
* This function returns the list of left braces of size <n>
*
***/
AllBraces := function(n)
local p, q, t, s, j, total;
t := Cputime();
p := [];
j := 0;
total := #{ x: x in {1..NumberOfSmallGroups(n)} | IsAbelian(SmallGroup(n,x)) };
for k in [1..NumberOfSmallGroups(n)] do
if not IsAbelian(SmallGroup(n,k)) then
continue;
end if;
j := j+1;
printf "id: %o,%o (%o/%o): ", n, k, j, total;
s := Cputime();
q := BracesWithAbelianGroup(SmallGroup(n,k));
if #q ne 0 then
for x in q do
Append(~p, x);
end for;
end if;
printf "%o found, %o seconds\n", #q, Cputime(s);
end for;
return p, Cputime(t);
end function;
