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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.
0
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.
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
KEYWORD
nonn
AUTHOR
Miles Englezou, Dec 27 2024
STATUS
approved