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

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 12, 1, 2, 1, 16, 1, 12, 1, 8, 3, 2, 1

Offset: 1

Author

Miles Englezou, Jul 14 2025

Keywords

• nonn
• more

Examples

			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.
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.
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.
		

Crossrefs

Cf. A340514.

Programs

GAP

# 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.
a := function(n, u, v)
    local T, S, k, r, m;
    T := [];
    for k in [1..NrSmallGroups(n)] do
        T := Concatenation(T, [SmallGroup(n,k)]);
    od;
    for m in [u..v] do
        S := [];
        for r in [1..NrSmallGroups(m*n)] do
            S := Concatenation(S, [SmallGroup(m*n, r)]);
        od;
        if ForAny(S, H -> ForAll(T, G -> ForAny(AllSubgroups(H), K -> IsomorphismGroups(G, K) <> fail))) then
            return m;
            break;
        fi;
    od;
    return fail;
end;

Formula

a(n) = A340514(n)/n.

a(p) = 1 for prime p.

a(p^2) = p.

a(p^3) = p^3 for p an odd prime.

If p < q are distinct primes, a(pq) = p if p divides (q-1), else a(pq) = 1.