cp's OEIS Frontend

A385999 Least k such that every group of order n embeds into a group of order k*n.

%I A385999 #11 Jul 19 2025 00:39:31
%S A385999 1,1,1,2,1,2,1,4,3,2,1,12,1,2,1,16,1,12,1,8,3,2,1
%N A385999 Least k such that every group of order n embeds into a group of order k*n.
%F A385999 a(n) = A340514(n)/n.
%F A385999 a(p) = 1 for prime p.
%F A385999 a(p^2) = p.
%F A385999 a(p^3) = p^3 for p an odd prime.
%F A385999 If p < q are distinct primes, a(pq) = p if p divides (q-1), else a(pq) = 1.
%e A385999 a(2) = 1 since there is one group of order 2 and therefore 2 is the least order such that all groups of order 2 are embedded, and 2/2 = 1.
%e A385999 a(4) = 2 since there are two groups of order 4 and both groups are embedded in a group of order 8, and 8/4 = 2.
%e A385999 a(12) = 12 since there are five groups of order 12 and 144 is the least order for which there is a group into which all five groups are embedded, and 144/12 = 12.
%o A385999 (GAP)
%o A385999 # Checks for n within the range [u..v]. In general u should be made equal to 1 to avoid erroneous output. Choice in range given for efficiency in checking individual terms.
%o A385999 a := function(n, u, v)
%o A385999     local T, S, k, r, m;
%o A385999     T := [];
%o A385999     for k in [1..NrSmallGroups(n)] do
%o A385999         T := Concatenation(T, [SmallGroup(n,k)]);
%o A385999     od;
%o A385999     for m in [u..v] do
%o A385999         S := [];
%o A385999         for r in [1..NrSmallGroups(m*n)] do
%o A385999             S := Concatenation(S, [SmallGroup(m*n, r)]);
%o A385999         od;
%o A385999         if ForAny(S, H -> ForAll(T, G -> ForAny(AllSubgroups(H), K -> IsomorphismGroups(G, K) <> fail))) then
%o A385999             return m;
%o A385999             break;
%o A385999         fi;
%o A385999     od;
%o A385999     return fail;
%o A385999 end;
%Y A385999 Cf. A340514.
%K A385999 nonn,more
%O A385999 1,4
%A A385999 _Miles Englezou_, Jul 14 2025