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Number of splitting-simple groups of order n; number of nontrivial groups of order n that are not semidirect products of proper subgroups.
2

%I #41 Nov 09 2024 22:53:37

%S 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,

%T 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,

%U 0,0,1,0,1,0,0,0,0,0,1,0,2,0,1,0,0,0,0,0,1

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

%C 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:

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

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

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

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

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

%C 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.

%C 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 ?

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

%C The conjecture that a(n) = 0 for nonprime squarefree n is true. Proof: It is known that every group G of squarefree order is supersolvable; hence G contains a normal series with prime cyclic factors. Since every Sylow subgroup of G is prime cyclic, these cyclic factors are isomorphic to the Sylow subgroups of G. Let P be one such factor; then for an appropriate M in G, P = G/M, where |G| = |P|*|M|. By the Schur-Zassenhaus theorem, G is a semidirect product of M and P, and a(n) = 0 when n is squarefree. - _Miles Englezou_, Oct 24 2024

%H Jianing Song, <a href="/A338757/b338757.txt">Table of n, a(n) for n = 1..255</a>

%H Tim Dokchitser, <a href="https://people.maths.bris.ac.uk/~matyd/GroupNames/extensions.html">Group extensions</a>

%H The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Splitting-simple_group">Splitting-simple group</a>

%H The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Permutable_complements">Permutable complements</a>

%H Jianing Song, <a href="/A338757/a338757_1.txt">List of splitting-simple groups of order up to 255 by ID in GAP's SmallGroup library</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F For primes p != q:

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

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

%F a(pq) = 0;

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

%F 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.

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

%F a(A120944(n)) = 0. - _Miles Englezou_, Oct 24 2024

%e a(48) = 1 because the binary octahedral group, which is of order 48, cannot be written as the semidirect product of proper subgroups.

%e a(16) = 2, and the corresponding groups are C_16 and Q_16 (generalized quaternion group of order 16).

%e a(81) = 2, and the corresponding groups are C_81 and SmallGroup(81,10).

%e 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.

%e 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]

%e 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.

%o (GAP)

%o IsSplittingSimple := function(G)

%o local c, l, i;

%o c := NormalSubgroups(G);

%o l := Length(c);

%o if l > 1 then

%o for i in [2..l-1] do

%o if Length(ComplementClassesRepresentatives(G,c[i])) > 0 then

%o return false;

%o fi;

%o od;

%o return true;

%o else

%o return false;

%o fi;

%o end;

%o A338757 := n -> Length(AllSmallGroups( n, IsSplittingSimple ));

%Y Cf. A000001, A090751 (number of directly indecomposable groups of order n), A001034, A120944.

%K nonn,hard

%O 1,8

%A _Jianing Song_, Nov 07 2020