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A137863
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Orders of simple groups which are non-cyclic and non-alternating.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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)
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REFERENCES
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L. E. Dickson, Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.
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LINKS
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EXAMPLE
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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)
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CROSSREFS
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Subsequence: A001228 (sporadic groups).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Incorrect formula and programs removed by R. J. Mathar, Apr 27 2020
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STATUS
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approved
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