A340515
a(n) = minimal order of a group in which all groups of order <= n can be embedded.
Original entry on oeis.org
1, 2, 6, 24, 120, 120, 840, 3360, 30240, 30240, 332640, 665280, 8648640, 8648640, 8648640
Offset: 1
- Heffernan, Robert, Des MacHale, and Brendan McCann. "Cayley's Theorem Revisited: Embeddings of Small Finite Groups." Mathematics Magazine 91.2 (2018): 103-111.
A340516
Let p_i (i=1..m) denote the primes <= n, and let e_i be the maximum number such that p_i^e_i <= n; then a(n) = Product_{i=1..m} p_i^(2*e_i-1).
Original entry on oeis.org
1, 2, 6, 24, 120, 120, 840, 3360, 30240, 30240, 332640, 332640, 4324320, 4324320, 4324320, 17297280, 294053760, 294053760, 5587021440, 5587021440, 5587021440, 5587021440, 128501493120, 128501493120, 3212537328000, 3212537328000, 28912835952000, 28912835952000, 838472242608000
Offset: 1
- Heffernan, Robert, Des MacHale, and Brendan McCann. "Cayley’s Theorem Revisited: Embeddings of Small Finite Groups." Mathematics Magazine 91.2 (2018): 103-111.
-
{1}~Join~Table[Times @@ Map[#^(2 Floor@ Log[#, n] - 1) &, Prime@ Range@ PrimePi@ n], {n, 2, 30}] (* Michael De Vlieger, Feb 23 2022 *)
A385999
Least k such that every group of order n embeds into a group of order k*n.
Original entry on oeis.org
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
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.
-
# 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;
Showing 1-3 of 3 results.
Comments