A377316 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.

8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 171, 172, 176, 180, 184, 188, 189

Offset

1,1

Comments

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.

The extension in question is by definition a central extension.

Links

Table of n, a(n) for n=1..57.

Formula

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).

Example

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.
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.

Prog

(GAP)
LoadPackage("sonata");
A:=[];
for n in [1..63] do
    for i in [1..NrSmallGroups(n)] do
        breakout:=false;
        G:=SmallGroup(n, i);
        Inn:=InnerAutomorphismsAutomorphismGroup(AutomorphismGroup(G));
        for k in [1..Length(AllSubgroups(G))] do
            if IsIsomorphicGroup(AllSubgroups(G)[k], Inn)=true
                and Order(Intersection(AllSubgroups(G)[k], Centre(G)))=1 then
                break;
            fi;
            if k=Length(AllSubgroups(G)) and
                IsIsomorphicGroup(AllSubgroups(G)[k], Inn)=false then
                A:=Concatenation(A, [n]);
                breakout:=true;
            fi;
        od;
        if breakout=true then
            break;
        fi;
    od;
od;
Print(A);

Crossrefs

Cf. A008586.

Sequence in context: A081925 A049199 A337806 * A192544 A302139 A160392

Adjacent sequences: A377313 A377314 A377315 * A377317 A377318 A377319

Keyword

nonn

Author

Miles Englezou, Dec 27 2024

Status

approved