A137863 Orders of simple groups which are non-cyclic and non-alternating.

168, 504, 660, 1092, 2448, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348, 175560, 178920, 194472, 246480, 262080

Offset: 1

Author

Artur Jasinski, Feb 16 2008

Keywords

• nonn

Comments

From Bernard Schott, Apr 26 2020: (Start)

About a(16) = 20160; 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but, 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8.

Indeed, 20160 is the smallest order for which there exist two nonisomorphic simple groups and it is the order of this group PSL_3(4) that was missing in the data. The first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1900) [see the link]. (End)

Examples

			From _Bernard Schott_, Apr 27 2020: (Start)
Two particular examples:
a(1) = 168 is the order of the smallest non-cyclic and non-alternating simple group, this Lie group is the projective special linear group PSL_2(7) that is isomorphic to the general linear group GL_3(2).
a(12) = 7920 is the order of the smallest sporadic group (A001228), the Mathieu group M_11. (End)
		

References

• L. E. Dickson, Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.

Links

• David Madore, Orders of non abelian simple groups
• Ida May Schottenfels, Two non isomorphic simple groups of the same order 20160, Annals of Mathematics, Second Series, Vol. 1, No. 1/4 (1900), pp. 147-152.

Crossrefs

Cf. A001034, A001710, A005180, A109379.

Subsequence: A001228 (sporadic groups).

Extensions

More terms from R. J. Mathar, Apr 23 2009

a(16) = 20160 inserted by Bernard Schott, Apr 26 2020

Incorrect formula and programs removed by R. J. Mathar, Apr 27 2020

Terms checked by Bernard Schott, Apr 26 2020