A001034 -id:A001034 - cp's OEIS frontend

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

Miles Englezou, Jun 15 2025

nonn

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.

			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.

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

Cf. A001034, A005180, A246655.

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;

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

A008976 Orders of non-cyclic simple groups (divided by 4).

Original entry on oeis.org

15, 42, 90, 126, 165, 273, 612, 630, 855, 1020, 1404, 1512, 1518, 1950, 1980, 2457, 3045, 3720, 5040, 6327, 6480, 7280, 8184, 8610, 9933, 12972, 14700, 15600, 18603, 23760, 25665, 28365, 31500, 37587, 43890, 44730

Offset: 1

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
M. Hall, Jr., A search for simple groups of order less than one million, pp. 137-168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory (page images), Dover, NY, 1958, p. 309.
David A. Madore, More terms
Index entries for sequences related to groups

Cf. A001034.

More terms from Sascha Kurz, Mar 24 2002

A145179 Number of finite noncyclic simple groups whose maximal order prime divisor is the n-th prime.

Original entry on oeis.org

0, 0, 3, 15, 10, 27, 18, 15, 14, 8, 28, 13, 17, 22, 10, 10, 3, 27, 11, 6, 34, 14, 9, 9, 6, 5, 14, 3, 12, 16, 24, 7, 12, 14, 4, 11, 17, 7, 7, 11, 4, 25, 9, 15, 3, 16, 20, 5, 3, 14, 11, 5, 25, 9, 51, 11, 4, 13, 6, 5, 13, 19, 16, 3, 17, 15, 32, 18, 3, 6, 10, 9, 10, 16, 5, 7, 9, 5, 11, 14, 3

Offset: 1

Andrei V. Zavarnitsine (zav(AT)math.nsc.ru), Oct 03 2008

nonn

			a(17)=3 because there are precisely three non-Abelian finite simple groups G (viz. PSL(2,59), A_59, A_60) such that the maximal prime divisor of the order of G is the 17th prime (which is 59).

A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum arXiv:081.0568 and Sib. Elec. Math. Rep. 6 (2009) 1-12

Cf. A001034

A189712 Numbers m such that for each prime p that divides m, there is a k(p) such that k(p) == 1 (mod p), k(p) divides m evenly, and m divides k(p)!/2 evenly.

Original entry on oeis.org

30, 56, 60, 90, 105, 120, 132, 144, 168, 180, 210, 240, 252, 264, 280, 288, 306, 315, 336, 351, 360, 380, 396, 400, 420, 432, 480, 495, 504, 520, 525, 528, 540, 546, 552, 560, 576, 612, 616, 630, 660, 672, 702, 720, 735, 756, 760, 792, 800, 810, 840, 858, 864, 900, 918, 924, 960, 990, 992, 1008, 1040, 1050, 1053, 1056, 1080, 1092, 1100, 1104

Offset: 1

James V. Blowers, Apr 25 2011

nonn

The order of every simple group must be a term of this sequence, so A001034 is a subsequence of this sequence. The Dahlke link shows the sequence with 72 and 112 (which are not in this sequence) added.

Karl Dahlke, Finite Groups, Simple Groups of Low Order

Cf. A001034.

findk(p, n) = {for (k = 1, n, if (((k % p) == 1) && ((n % k) == 0) && ((k! % 2) == 0) && (((k!/2) % n) == 0), return (1));); return (0);}
isok(n) = {vp = factor(n)[,1]~; for (i = 1, #vp, if (! findk(vp[i], n), return (0));); return (1);} \\ Michel Marcus, Aug 22 2013

A327912 Orders of perfect non-simple groups.

Original entry on oeis.org

120, 336, 720, 960, 1080, 1320, 1344, 1920, 2160, 2184, 2688, 3000, 3600, 3840, 4860, 4896, 5040, 5376, 5760, 6840, 7200, 7500, 7560, 7680, 9720, 10080, 10752, 11520, 12144, 14400, 14520, 14580, 15000, 15120, 15360, 15600, 16464, 17280, 19656, 20160, 21504, 21600, 23040, 24360, 28224, 29160, 29760, 30240, 30720, 32256, 34560, 37500, 39600, 40320, 43008, 43200, 43320, 43740, 46080, 48000, 50616, 51840, 56448, 57600, 57624, 58240, 58320, 60480

Offset: 1

Sébastien Palcoux, Sep 29 2019

nonn

The smallest number n such that there is a simple group and a non-simple perfect group of order n is 20160. So this sequence is A060793 minus A001034 (as sets) for the orders less than 20160. The next known such exceptions are 181440, 262080, 443520 and 604800.

The perfect groups of order 61440, 122880, 172032, 245760, 344064, 491520, 688128, 983040 have not completely been determined yet. Then GAP neither provides the number of these groups nor the groups themselves.

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.
D.F. Holt and W. Plesken, Perfect Groups, Oxford Math. Monographs, Oxford University Press, 1989.

The GAP Group, GAP Data Library "Perfect Groups".

Cf. A001034, A005180, A060793.

OrderPerfectNonSimple:=function(n1,n2)
   local it,S,G,L,o,No,i,c;
   it:=SimpleGroupsIterator(n1,n2);
   S:=[];
   for G in it do
      Add(S,Order(G));
   od;
   L:=[];
   for o in [n1..n2] do
      c:=0;
      for i in S do
         if i=o then
            c:=c+1;
         fi;
      od;
      No:=NumberPerfectGroups(o);
      if No>c then
         Add(L,o);
         if c>0 then
            Print([o,c,No]);
         fi;
      fi;
   od;
   return L;
end;;

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A385030 Orders of characteristically simple groups.

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A008976 Orders of non-cyclic simple groups (divided by 4).

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A145179 Number of finite noncyclic simple groups whose maximal order prime divisor is the n-th prime.

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A189712 Numbers m such that for each prime p that divides m, there is a k(p) such that k(p) == 1 (mod p), k(p) divides m evenly, and m divides k(p)!/2 evenly.

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A327912 Orders of perfect non-simple groups.

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