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A385030 Orders of characteristically simple groups.

%I A385030 #16 Jun 18 2025 18:04:42
%S A385030 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,
%T A385030 60,61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,127,128,
%U A385030 131,137,139,149,151,157,163,167,168,169,173,179,181,191,193,197,199,211
%N A385030 Orders of characteristically simple groups.
%C A385030 Equivalently, orders k of groups G where a G exists as a direct product of isomorphic simple groups.
%C A385030 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.
%H A385030 Miles Englezou, <a href="/A385030/b385030.txt">Table of n, a(n) for n = 1..10000</a>
%H A385030 Wikipedia, <a href="https://en.wikipedia.org/wiki/Characteristically_simple_group">Characteristically simple group</a>
%F A385030 Union of A246655 and the nonzero powers of every term in A001034.
%e A385030 5 is a term since C_5 is prime cyclic and contains no proper subgroups. Therefore it contains no characteristic proper subgroups.
%e A385030 60 is a term since the alternating group A_5 is simple and contains no normal subgroups. Therefore it contains no characteristic proper subgroups.
%e A385030 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.
%o A385030 (GAP)
%o A385030 isok := function(G)
%o A385030     if Order(G) = 1 then
%o A385030         return false;
%o A385030     elif IsElementaryAbelian(G) then
%o A385030         return true;
%o A385030     elif IsSimpleGroup(G) then
%o A385030         return true;
%o A385030     else
%o A385030         for K in AllSubgroups(G) do
%o A385030             if IsCharacteristicSubgroup(G, K) then
%o A385030                 return false;
%o A385030             fi;
%o A385030         od;
%o A385030         return true;
%o A385030     fi;
%o A385030 end;
%Y A385030 Cf. A001034, A005180, A246655.
%K A385030 nonn
%O A385030 1,1
%A A385030 _Miles Englezou_, Jun 15 2025