A119648 Orders for which there is more than one simple group.

20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480

Offset

1,1

Comments

All such orders are composite numbers (since there is only one group of any prime order).

Orders which are repeated in A109379.

Except for the first number, these are the orders of symplectic groups C_n(q)=Sp_{2n}(q), where n>2 and q is a power of an odd prime number (q=3,5,7,9,11,...). Also these are the orders of orthogonal groups B_n(q). - Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

a(1) = 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 (see A137863). - Bernard Schott, May 18 2020

References

See A001034 for references and other links.

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]. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Links

Table of n, a(n) for n=1..10.

C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. See also. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

W. Kimmerle et al., Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups, Proc. London Math. Soc., (3) 60 (1990), 89-122. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

David A. Madore, Orders of non-Abelian simple groups

Wikipedia, Classification of finite simple groups [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Index entries for sequences related to groups

Formula

For n>1, a(n) is obtained as (1/2) q^(m^2)Prod(q^(2i)-1, i=1..m) for appropriate m>2 and q equal to a power of some odd prime number. [Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Example

From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: (Start)
a(1)=|A_8|=8!/2=20160,
a(2)=|C_3(3)|=4585351680,
a(3)=|C_3(5)|=228501000000000, and
a(4)=|C_4(3)|=65784756654489600. (End)

Prog

(Other) sp(n, q) 1/2 q^n^2.(q^(2.i) - 1, i, 1, n) [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] [This line contained some nonascii characters which were unreadable]

Crossrefs

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders of simple groups without repetition), A109379 (orders with repetition), A137863 (orders of simple groups which are non-cyclic and non-alternating).

Sequence in context: A305186 A181233 A262776 * A127224 A235523 A235520

Adjacent sequences: A119645 A119646 A119647 * A119649 A119650 A119651

Keyword

nonn

Author

N. J. A. Sloane, Jul 29 2006

Extensions

Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

Status

approved