OFFSET
1,1
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.
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.
EXAMPLE
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).
PROG
(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);
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Miles Englezou, Jan 01 2025
STATUS
approved