cp's OEIS Frontend

A376065 Orders k of groups G such that Inn(G) is isomorphic to Z(Aut(G)) for at least one G of order k.

%I A376065 #39 Sep 16 2024 09:22:44
%S A376065 1,2,4,8,16,24,32,48,64,72,96,128,144,192,216,243,256,288,320,384,432,
%T A376065 486,512,576,640,648,729,768,864
%N A376065 Orders k of groups G such that Inn(G) is isomorphic to Z(Aut(G)) for at least one G of order k.
%C A376065 Inn(G) is the inner automorphism group of G and Z(Aut(G)) is the center of the automorphism group of G.
%C A376065 A group G for which Inn(G) = Z(Aut(G)) allows for a natural construction of Aut(Aut(G)) via the homomorphism f: Aut(G) -> Aut(Aut(G)) which maps Aut(G) to Inn(Aut(G)) = Aut(G)/Z(Aut(G)) in the same way that G is mapped to Inn(G) = G/Z(G). Furthermore Inn(Aut(G)) = Out(G) (the outer automorphism group), and we have an exact sequence of homomorphisms 1 -> G -> Aut(G) -> Aut(Aut(G)) -> 1. Each term a(n) is thus the order of a group which allows for this particular construction of Aut(Aut(G)).
%C A376065 The diagram of homomorphisms is as follows:
%C A376065              Aut(Aut(G)) --> Out(Aut(G))
%C A376065             /      ^          /
%C A376065            /       |         /
%C A376065      Aut(G)  -->  Inn(Aut(G)) (= Aut(G)/Z(Aut(G)) = Out(G))
%C A376065     /  ^         /
%C A376065    /   |        /
%C A376065  G --> Z(Aut(G)) (= Inn(G))
%C A376065  ^    /
%C A376065  |   /
%C A376065  Z(G)
%C A376065 A000079, A007283(m) for m >= 3, and A020714(r) for r >= 6, are subsequences. See the Miles Englezou link for proofs. In the link it is also shown that the method of proof used to determine that A007283(m) and A020714(r) are subsequences is limited to Fermat primes (A019434) and therefore cannot be used to determine whether 2^s*p is a subsequence for every prime p.
%H A376065 Miles Englezou, <a href="/A376065/a376065_4.txt">Proofs of subsequences</a>
%e A376065 24 is a term since for G = C3 x D8, Inn(G) = Z(Aut(G)) = C2 x C2, and G has order 24.
%o A376065 (GAP)
%o A376065 S:=[];
%o A376065 for n in [1..500] do
%o A376065     for i in [1..NrSmallGroups(n)] do
%o A376065         G:=SmallGroup(n,i);
%o A376065         Aut:=AutomorphismGroup(G);
%o A376065         Inn:=InnerAutomorphismsAutomorphismGroup(Aut);
%o A376065         if IsomorphismGroups(Centre(Aut),Inn)<>fail then
%o A376065             S:=Concatenation(S,[n]);
%o A376065             break;
%o A376065         fi;
%o A376065     od;
%o A376065 od;
%o A376065 Print(S);
%Y A376065 Cf. A000079, A007283, A020714, A019434.
%K A376065 nonn,more
%O A376065 1,2
%A A376065 _Miles Englezou_, Sep 07 2024