OFFSET
1,8
COMMENTS
There are a few equivalent conditions for a group G to be a semidirect product of subgroups N and H, where N is normal in G (if H is also normal, then the semidirect product is a direct product). See the Wikipedia link for these equivalent conditions.
One condition with an interesting consequence is the following. If G is a semidirect product of N and H, then the exact sequence of homomorphisms 1 -> N -> G -> H -> 1 is split, meaning there exists a homomorphism f: H -> G. When a sequence is split, H = G/N is isomorphic to the image of f, which is a subgroup of G. This means that the extension factors which form G are contained within G (up to isomorphism). For a sequence 1 -> A -> G -> B -> 1 which is not split (and therefore G is not a semidirect product) the factor B = G/A is not isomorphic to a subgroup of G, and therefore G, seen as an extension of B by A, is composed of a factor B which lies outside of it.
LINKS
Wikipedia, Semidirect product
FORMULA
a(A000040(n)) = 1 since the only group of prime order is prime cyclic which is trivially not a semidirect product as it contains no proper subgroups let alone proper normal subgroups.
a(A120944(n)) = 0. This is because every group of squarefree order contains a normal Hall subgroup (a subgroup whose order is coprime to its index). Every such group is a semidirect product by the Schur-Zassenhaus theorem.
a(p^a*q^b) = 0 for distinct primes p and q, a >= 1, b >= 1. This is also because every G of order p^a*q^b contains a normal Hall subgroup and is therefore a semidirect product by the Schur-Zassenhaus theorem.
EXAMPLE
a(2) = 1 since the cyclic group C2 is not a semidirect product and is the only group of order 2.
a(6) = 0 since C6 and S3, the symmetric group on 3 elements, are both semidirect products and are the only groups of order 6. (C6 = C2 x C3, and S3 = D6 = C2 : C3).
a(8) = 2 since C8 and Q8, the quaternion group, are not semidirect products and every other group of order 8 is a semidirect product.
PROG
(GAP)
M:=[]; U:=[]; LoadPackage("sonata");
for n in [1..63] do
M:=Concatenation(M, [NrSmallGroups(n)]);
S:=[];
for i in [1..NrSmallGroups(n)] do
breakout:=false;
G:=SmallGroup(n, i);
T:=[];
for j in [1..Size(Subgroups(G))] do
for k in [1..Size(Subgroups(G))] do
A:=Subgroups(G)[j]; B:=Subgroups(G)[k];
if Size(A) = 1 then
continue;
fi;
if G <> A and IsNormal(G, A)=true and Size(Intersection(A, B))= 1
and G=Group(Union(GeneratorsOfGroup(A), GeneratorsOfGroup(B)))
then
T:=Concatenation(T, [i]);
S:=Concatenation(S, [T]);
breakout:=true;
break;
fi;
od;
if breakout=true then
break;
fi;
od;
od;
U:=Concatenation(U, [Size(S)]);
od;
Print(M-U);
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Miles Englezou, Oct 08 2024
STATUS
approved