A385030 Orders of characteristically simple groups.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 168, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Author

Miles Englezou, Jun 15 2025

Keywords

• nonn

Comments

Equivalently, orders k of groups G where a G exists as a direct product of isomorphic simple groups.

A group G is characteristically simple if it contains no characteristic proper subgroups (a subgroup which is invariant under every automorphism of G). Since a finite group is characteristically simple if and only if it is a direct product of isomorphic simple groups, G is characteristically simple if and only if it is an elementary abelian group or a direct product of isomorphic nonabelian simple groups.

Examples

			5 is a term since C_5 is prime cyclic and contains no proper subgroups. Therefore it contains no characteristic proper subgroups.
60 is a term since the alternating group A_5 is simple and contains no normal subgroups. Therefore it contains no characteristic proper subgroups.
3600 is a term since the direct product A_5 x A_5, though it contains A_5 twice as a normal subgroup and is therefore not simple, it contains no characteristic proper subgroups.
		

Links

• Miles Englezou, Table of n, a(n) for n = 1..10000
• Wikipedia, Characteristically simple group

Crossrefs

Cf. A001034, A005180, A246655.

Programs

GAP

isok := function(G)
    if Order(G) = 1 then
        return false;
    elif IsElementaryAbelian(G) then
        return true;
    elif IsSimpleGroup(G) then
        return true;
    else
        for K in AllSubgroups(G) do
            if IsCharacteristicSubgroup(G, K) then
                return false;
            fi;
        od;
        return true;
    fi;
end;

Formula

Union of A246655 and the nonzero powers of every term in A001034.