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A379755 Orders k of groups where at least one group has a simple automorphism group.

%I A379755 #9 Jan 14 2025 09:54:09
%S A379755 3,4,6,8,16,32,64,128,256,512,1024
%N A379755 Orders k of groups where at least one group has a simple automorphism group.
%C A379755 A group G that has a simple automorphism group Aut(G) is either abelian or simple and complete (that is, the center Z(G) = 1 and G = Aut(G)). Proof: since the group of inner automorphisms Inn(G) is a normal subgroup of Aut(G), if Aut(G) is simple then Inn(G) = 1 or Aut(G). When Inn(G) = 1, G is abelian. That G is simple and complete when Inn(G) = Aut(G) can be proved by considering that the homomorphism f: G -> Aut(G), with Inn(G) as image and Z(G) as kernel, is surjective, and that a surjective homomorphism preserves normal subgroups. As Aut(G) is simple, therefore G must also be simple, otherwise any normal N in G would correspond to a normal f(N) in Aut(G). Since G is nonabelian, Z(G) = 1, and Aut(G) = Inn(G) = G/Z(G) = G. Hence, when Aut(G) is simple and Inn(G) = Aut(G), G is simple and complete.
%C A379755 As the order of any simple complete group is a term, the orders of various sporadic simple groups are also terms, including 7920, 10200960, and 244823040, corresponding to the Mathieu groups M_11, M_23, and M_24, and also 175560, corresponding to the Janko group J_1.
%F A379755 2^m is a term for m >= 3. This is because every elementary abelian group of order 2^m has the projective special linear group PSL(m,2) as an automorphism group, which is simple for every such m.
%e A379755 3 is a term since Aut(C_3) = C_2, which is simple.
%e A379755 8 is a term since Aut(C_2 x C_2 x C_2) = PSL(3,2), which is simple.
%e A379755 7920 is a term since Aut(M_11) = M_11, which is simple. (M_11 is one of the Mathieu groups).
%o A379755 (GAP)
%o A379755 A:=[];
%o A379755 for n in [1..100] do
%o A379755     if n>=8 and Unique(Factors(n))=[2] then
%o A379755         A:=Concatenation(A,[n]);
%o A379755         continue;
%o A379755     fi;
%o A379755     for i in [1..NrSmallGroups(n)] do
%o A379755         G:=SmallGroup(n,i);
%o A379755         if IsAbelian(G)=false and Order(Centre(G))>1 then
%o A379755             continue;
%o A379755         fi;
%o A379755         Aut:=AutomorphismGroup(G);
%o A379755         if IsSimpleGroup(Aut)=true then
%o A379755             A:=Concatenation(A,[n]);
%o A379755             break;
%o A379755         fi;
%o A379755     od;
%o A379755 od;
%o A379755 Print(A);
%Y A379755 Cf. A000079, A001228.
%K A379755 nonn,more
%O A379755 1,1
%A A379755 _Miles Englezou_, Jan 01 2025