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%I A335419 #33 Jul 14 2020 07:03:13
%S A335419 1,4,6,8,9,10,12,14,15,16,18,21,20,22,24,25,26,27,28,30,32,33,34,35,
%T A335419 36,38,39,40,42,44,45,46,48,49,50,51,52,54,55,56,57,58,62,63,64,65,66,
%U A335419 68,69,70,72,74,75,76,77,78,80,81,82,84,85,86,87,88,90,91,92,93,94
%N A335419 Integers m such that every group of order m is not simple.
%C A335419 Officially, the group of order 1 is not considered to be simple; "a group <> 1 is simple if it has no normal subgroups other than G and 1" (See reference for Joseph J. Rotman's definition).
%C A335419 There is no prime term because there exists only one group of order p and this cyclic group Z/pZ is simple.
%C A335419 As a consequence of Feit-Thompson theorem, all odd composites are terms of this sequence.
%C A335419 The first composite even number that is not present in the data is 60 that is the order of simple alternating group Alt(5), the second one that is missing is 168 corresponding to simple Lie group PSL(3,2) [A031963].
%D A335419 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, Exercice 1.44 p.96.
%D A335419 Joseph J. Rotman, The Theory of Groups: An Introduction, 4th ed., Springer-Verlag, New-York, 1995. Page 39, Definition.
%H A335419 Craig Cato, <a href="https://doi.org/10.1090/S0025-5718-1977-0430052-X">The orders of the known simple groups as far as one trillion</a>, Math. Comp., 31 (1977), 574-577.
%H A335419 Wikipedia, <a href="https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson_theorem">Feit-Thompson theorem</a>.
%e A335419 There exist 5 (nonisomorphic) groups of order 8: Z/8Z, Z/2Z × Z/4Z, (Z/2Z)^3, D_4 and H_8; none of these 5 groups is simple, so 8 is a term.
%e A335419 There exist 13 (nonisomorphic) groups of order 60 (see A000001), 12 are not simple but the alternating group Alt(5) is simple, hence 60 is not a term.
%Y A335419 Complement of A005180 (except for 1).
%Y A335419 Subsequence: A014076 (odd nonprimes).
%Y A335419 Cf. A000001, A031963, A051532 (similar for Abelian), A056867 (similar for nilpotent).
%K A335419 nonn
%O A335419 1,2
%A A335419 _Bernard Schott_, Jul 09 2020

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