A338757 Number of splitting-simple groups of order n; number of nontrivial groups of order n that are not semidirect products of proper subgroups.

0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 5, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 19, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1

Offset

1,8

Comments

The other names for groups of this kind include "semidirectly indecomposable groups" or "inseparable groups". Note that the following are equivalent definitions for a nontrivial group to be a splitting-simple group:

- It is not the (internal) semidirect product of proper subgroups;

- It is not isomorphic to the (external) semidirect product of nontrivial groups;

- It has no proper nontrivial normal subgroups with a permutable complement.

- It is the non-split extension of every proper nontrivial normal subgroup by the corresponding quotient group.

Also note that being simple is a stronger condition than being splitting-simple, while being directly indecomposable (see A090751) is weaker.

a(p^e) >= 1 since C_p^e cannot be written as the semidirect product of proper subgroups. For e >= 3, a(2^e) >= 2 by the existence of the generalized quaternion group of order 2^e, which is the only non-split extension of C_2^(e-1) by C_2 other than C_2^e.

The smallest numbers here with a(n) > 0 that are not prime powers are 48, 60, 120, 144, 168, 192, 240, 320, 336, 360 and so on. Are there any odd numbers n that are not prime powers satisfying a(n) > 0 ?

Conjecture: a(n) = 0 for squarefree n which is not a prime.

Links

Jianing Song, Table of n, a(n) for n = 1..255

Tim Dokchitser, Group extensions

The Group Properties Wiki, Splitting-simple group

The Group Properties Wiki, Permutable complements

Jianing Song, List of splitting-simple groups of order up to 255 by ID in GAP's SmallGroup library

Index entries for sequences related to groups

Formula

For primes p != q:

a(p) = a(p^2) = 1; a(p^3) = 2 for p = 2, 1 otherwise;

a(p^4) = 2 for p = 2 or 3, 1 otherwise;

a(pq) = 0;

a(4p) = a(8p) = 0, p > 2.

a(n) <= A090751(n) for all n, and the equality holds if n = 1, p, p^2 for primes p or n = pq for primes p < q and p does not divide q-1.

a(A001034(k)) >= 1, since A001034 lists the orders of (non-Abelian) simple groups.

Example

a(48) = 1 because the binary octahedral group, which is of order 48, cannot be written as the semidirect product of proper subgroups.
a(16) = 2, and the corresponding groups are C_16 and Q_16 (generalized quaternion group of order 16).
a(81) = 2, and the corresponding groups are C_81 and SmallGroup(81,10).
a(64) = 19, and the corresponding groups are SmallGroup(64,i) for i = 1, 11, 13, 14, 19, 22, 37, 43, 45, 49, 54, 79, 81, 82, 160, 168, 172, 180 and 245.
For n = 60 or 168, the unique simple group is the only group of order n that cannot be written as the semidirect product of proper subgroups, hence a(60) = a(168) = 1. [The unique simple groups are respectively Alt(5) and PSL(2,7). - Bernard Schott, Nov 08 2020]
For n = 12, we have C_12 = C_3 X C_4, C_6 X C_2 = C_6 X C_2, D_6 = C_6 : C_2, Dic_12 = C_3 : C_4 and A_4 = (C_2 X C_2) : C_3, all of which can be written as the semidirect product of nontrivial groups. So a(12) = 0.

Prog

(GAP)
IsSplittingSimple := function(G)
  local c, l, i;
  c := NormalSubgroups(G);
  l := Length(c);
  if l > 1 then
    for i in [2..l-1] do
    if Length(ComplementClassesRepresentatives(G, c[i])) > 0 then
      return false;
    fi;
    od;
    return true;
  else
    return false;
  fi;
end;
A338757 := n -> Length(AllSmallGroups( n, IsSplittingSimple ));

Crossrefs

Cf. A000001, A090751 (number of directly indecomposable groups of order n), A001034.

Sequence in context: A144627 A135929 A346377 * A080733 A080732 A301295

Adjacent sequences: A338754 A338755 A338756 * A338758 A338759 A338760

Keyword

nonn,hard

Author

Jianing Song, Nov 07 2020

Status

approved