A338853 List of numbers k > 1 such that there exists a group of order k without nontrivial normal Sylow subgroups.

24, 48, 60, 72, 96, 120, 144, 160, 168, 180, 192, 216, 240, 288, 300, 320, 324, 336, 360, 384, 432, 480, 504, 540, 576, 600, 640, 648, 660, 672, 720, 768, 784, 800, 840, 864, 896, 900, 960, 972, 1008, 1053, 1080, 1092, 1152, 1176, 1200, 1280, 1296, 1320, 1344, 1440

Offset: 1

Author

Jianing Song, Nov 12 2020

Keywords

• nonn

Comments

Equivalently, numbers k > 1 such that there exists a group of order k with Sylow number > 1 for every prime dividing k.

The corresponding numbers of groups of order k without nontrivial normal Sylow subgroups are: 1, 4, 1, 4, 17, 3, 17, 1, 1, 1, 86, 18, 8, 90, 1, 5, 1, 3, 6, 536, 80, 27, ...

Note that if G has no normal Sylow p-subgroups, p divides |G|, then G X C_p also has no normal Sylow p-subgroups. That is to say, if k is in this sequence and p divides k, then k*p is also in this sequence. In particular, every number of the form 24 * 2^a * 3^b * 5^c * 7^d with nonnegative a,b,c,d is here.

The "primitive" terms (the terms not of the form k*p where k is a previous term and p divides k) are 24, 60, 160, 168, 324, 660, 784, 840, 896, 1053, ...

Includes A001034 as a subsequence, by the definition of non-cyclic simple groups. If q is not a prime power (not in A246655) and A338757(q) > 0, then q is here, as guaranteed by Schur-Zassenhaus theorem.

If k = p^e * q is here, p, q distinct primes, then q == 1 (mod p) and e >= 1+ord(p,q), where ord(p,q) is the multiplicative order of p modulo q. Proof: Let G be a group of order k without nontrivial normal Sylow subgroups. Let n_p (respectively n_q) be the number of Sylow p-subgroups (respectively q-subgroups), then n_p, n_q > 1. By Sylow's 3rd theorem, we have n_p == 1 (mod p), n_p | q; n_q == 1 (mod q), n_q | p^e. It is possible only if q == 1 (mod p) and e >= ord(p,q).

If e = ord(p,q), then we must have n_q = p^e. The Sylow q-subgroups have order q, which is a prime, so the pairwise intersections must be trivial, i.e., there are p^e * (q-1) = k - p^e elements in G of order q. The remaining p^e elements are just enough to make a unique Sylow p-subgroup, so n_p = 1, which is a contradiction. Hence, e >= 1+ord(p,q).

The terms of the form p^e * q where e = 1+ord(p,q), q == 1 (mod p) are 24 = 2^3 * 3, 160 = 2^5 * 5, 1053 = 3^4 * 13 and so on. Note that q == 1 (mod p) and e >= 1+ord(p,q) are only necessary but not sufficient: 112 = 2^4 * 7 satisfies 7 == 1 (mod 2) and 4 >= 1+ord(2,7), but 112 is not here. Similarly, 19375 = 5^4 * 31 satisfies 31 == 1 (mod 5) and 4 >= 1+ord(5,31), but 19375 is not here.

Examples

			All the normal subgroups of S_4 (symmetric group of degree 4, order 24) are the trivial group, the Klein four-group (order 4), A_4 (alternating group of degree 4, order 12) and S_4 itself. None of these is a Sylow 2-subgroup or a Sylow 3-subgroup. So 24 is a term.
All the normal subgroups of SmallGroup(1053,51) are the trivial group, C_3 X C_3 X C_3 (order 27), SmallGroup(351,12) and SmallGroup(1053,51) itself. None of these is a Sylow 3-subgroup or a Sylow 13-subgroup. So 1053 is a term. In fact, 1053 is the smallest odd term. [As a result, every number of the form 1053 * 3^a * 13^b with nonnegative a,b is a term, showing that there are infinitely many odd terms in this sequence. What is the smallest odd term not of this form? - _Jianing Song_, Sep 08 2021]
		

Links

• Jianing Song, Table of n, a(n) for n = 1..66
• The Group Properties Wiki, Sylow subgroup
• The Group Properties Wiki, Schur-Zassenhaus theorem
• Index entries for sequences related to groups

Crossrefs

Cf. A001034, A246655, A338757.

Programs

GAP

HasNoSylow := function(G)
  local c, l, i;
  c := FactInt(Size(G))[1];
  l := Length(c);
  if c[1] = c[l] then     # |G| is 1 or a prime power
    return false;
  else
    for i in [1..l] do
      if IsNormal(G, SylowSubgroup(G, c[i])) then
        return false;
      fi;
    od;
    return true;
  fi;
end;
IsA338853 := function(n)
  local c, l, i;
  c := FactInt(n)[1];
  l := Length(c);
  if c[1] = c[l] then     # |G| is 1 or a prime power
    return false;
  else
    i := NumberSmallGroups(n);
    while i > 0 do
      if(HasNoSylow(SmallGroup(n,i))) then
        return true;
      fi;
      i := i-1;
    od;
    return false;
  fi;
end;