A335419 Integers m such that every group of order m is not simple.

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Author

Bernard Schott, Jul 09 2020

Keywords

• nonn

Comments

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).

There is no prime term because there exists only one group of order p and this cyclic group Z/pZ is simple.

As a consequence of Feit-Thompson theorem, all odd composites are terms of this sequence.

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].

Examples

			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.
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.
		

References

• Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, Exercice 1.44 p.96.
• Joseph J. Rotman, The Theory of Groups: An Introduction, 4th ed., Springer-Verlag, New-York, 1995. Page 39, Definition.

Links

• Craig Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
• Wikipedia, Feit-Thompson theorem.

Crossrefs

Complement of A005180 (except for 1).

Subsequence: A014076 (odd nonprimes).

Cf. A000001, A031963, A051532 (similar for Abelian), A056867 (similar for nilpotent).