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

3, 4, 6, 8, 16, 32, 64, 128, 256, 512, 1024

Offset: 1

Author

Miles Englezou, Jan 01 2025

Keywords

• nonn
• more

Comments

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.

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.

Examples

			3 is a term since Aut(C_3) = C_2, which is simple.
8 is a term since Aut(C_2 x C_2 x C_2) = PSL(3,2), which is simple.
7920 is a term since Aut(M_11) = M_11, which is simple. (M_11 is one of the Mathieu groups).
		

Crossrefs

Cf. A000079, A001228.

Programs

GAP

A:=[];
for n in [1..100] do
    if n>=8 and Unique(Factors(n))=[2] then
        A:=Concatenation(A,[n]);
        continue;
    fi;
    for i in [1..NrSmallGroups(n)] do
        G:=SmallGroup(n,i);
        if IsAbelian(G)=false and Order(Centre(G))>1 then
            continue;
        fi;
        Aut:=AutomorphismGroup(G);
        if IsSimpleGroup(Aut)=true then
            A:=Concatenation(A,[n]);
            break;
        fi;
    od;
od;
Print(A);

Formula

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.