A377316 - OEIS

%I #7 Jan 07 2025 20:27:40

%S 8,12,16,20,24,27,28,32,36,40,44,48,52,54,56,60,63,64,68,72,76,80,81,

%T 84,88,92,96,100,104,108,112,116,117,120,124,125,126,128,132,135,136,

%U 140,144,148,152,156,160,162,164,168,171,172,176,180,184,188,189

%N Orders k of groups G such that G is a non-split extension of Inn(G) by Z(G) for at least one group G of order k.

%C Inn(G) being the group of inner automorphisms of G and Z(G) the center of G, since Inn(G) = G/Z(G) every group G can be seen as an extension of Inn(G) by Z(G) (considering Inn(G) as a proper group isomorphic to the quotient G/Z(G)). If Inn(G) is isomorphic to a subgroup H of G and the intersection of H and Z(G) is trivial then the extension splits: intuitively this means that G is composed of extension factors that are contained within G itself. If Inn(G) = K and K is not a subgroup of G, or if it is then if the intersection of K and Z(G) is nontrivial, then the extension does not split, meaning that in this particular construction G is composed of a factor K which lies outside of it.

%C The extension in question is by definition a central extension.

%F 4*m is a term for m > 1. This can be proved by considering the dihedral group D_4m: for every such group Z(D_4m) = C_2 and Inn(D_4m) = D_2m; both are subgroups of D_4m up to isomorphism, however the intersection of D_2m and C_2 is not trivial, and therefore D_4m is a non-split extension of Inn(D_4m) by Z(D_4m).

%e 8 is a term since for the quaternion group Q_8, Inn(Q_8) = C_2 x C_2 and C_2 x C_2 is not a subgroup of Q_8.

%e 27 is a term since for the group (C_3 x C_3) : C_3 (':' denoting the semidirect product) Inn((C_3 x C_3) : C_3) = C_3 x C_3, and although C_3 x C_3 is a subgroup of (C_3 x C_3) : C_3, the intersection of C_3 x C_3 and Z(C_3 x C_3) is not trivial.

%o (GAP)

%o LoadPackage("sonata");

%o A:=[];

%o for n in [1..63] do

%o     for i in [1..NrSmallGroups(n)] do

%o         breakout:=false;

%o         G:=SmallGroup(n,i);

%o         Inn:=InnerAutomorphismsAutomorphismGroup(AutomorphismGroup(G));

%o         for k in [1..Length(AllSubgroups(G))] do

%o             if IsIsomorphicGroup(AllSubgroups(G)[k],Inn)=true

%o                 and Order(Intersection(AllSubgroups(G)[k],Centre(G)))=1 then

%o                 break;

%o             fi;

%o             if k=Length(AllSubgroups(G)) and

%o                 IsIsomorphicGroup(AllSubgroups(G)[k],Inn)=false then

%o                 A:=Concatenation(A,[n]);

%o                 breakout:=true;

%o             fi;

%o         od;

%o         if breakout=true then

%o             break;

%o         fi;

%o     od;

%o od;

%o Print(A);

%Y Cf. A008586.

%K nonn

%O 1,1

%A _Miles Englezou_, Dec 27 2024