A109379 -id:A109379 - cp's OEIS frontend

A001034 Orders of noncyclic simple groups (without repetition).

60, 168, 360, 504, 660, 1092, 2448, 2520, 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

Offset: 1

N. J. A. Sloane, Simon Plouffe

An alternative definition, to assist in searching: Orders of non-cyclic finite simple groups.
This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
The primitive elements are A257146. These are also the primitive elements of A056866. - Charles R Greathouse IV, Jan 19 2017
Conjecture: This is a subsequence of A083207 (Zumkeller numbers). Verified for n <= 156. A fast provisional test was used, based on Proposition 17 from Rao/Peng JNT paper (see A083207). For those few cases where the fast test failed (such as 2588772 and 11332452) the comprehensive (but much slower) test by T. D. Noe at A083207 was used for result confirmation. - Ivan N. Ianakiev, Jan 11 2020
From M. Farrokhi D. G., Aug 11 2020: (Start)
The conjecture is not true. The smallest and the only counterexample among the first 457 terms of the sequence is a(175) = 138297600.
On the other hand, the orders of sporadic simple groups are Zumkeller. And with the exception of the smallest two orders 7920 and 95040, the odd part of the other orders are also Zumkeller. (End)
Every term in this sequence is divisible by 4*p*q, where p and q are distinct odd primes. - Isaac Saffold, Oct 24 2021

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].
Dickson L.E. Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.
M. Hall, Jr., A search for simple groups of order less than one million, pp. 137-168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000
C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory (page images), Dover, NY, 1958, p. 309.
M. Farrokhi D. G., A GAP function generating the smallest non-cyclic finite simple group order greater than a given number m.
Walter Feit and J. G. Thompson, A solvability criterion for finite groups and some consequences, Proc. N. A. S. 48 (6) (1962) 968.
M. Hall Jr., Simple groups of order less than one million, J. Alg. 20 (1) (1972) 98-102
David A. Madore, More terms
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
Index entries for sequences related to groups
Index entries for "core" sequences

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

Cf. A109379 (orders with repetition), A119648 (orders that are repeated).

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 29 2006

nonn

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

			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)

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]

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

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).

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]

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]

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

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

Original entry on oeis.org

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

Artur Jasinski, Feb 16 2008

nonn

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)

			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)

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

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.

Cf. A001034, A001710, A005180, A109379.

Subsequence: A001228 (sporadic groups).

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

A330583 The orders, with repetition, of the non-cyclic finite simple groups whose orders are 23-smooth.

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 20160, 20160, 25920, 29120, 58800, 62400, 95040, 126000, 175560, 181440, 262080, 443520, 604800, 979200, 1451520, 1814400

Offset: 1

Hal M. Switkay, Dec 18 2019

nonn

This is the intersection of A109379 and A080683. It should be a finite set; a proof thereof would be welcome.

			This list contains the orders of all non-cyclic finite simple groups < 12180. However, 29|12180, which is the order of L2(29).

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].

Hal M. Switkay, Table of n, a(n) for n = 1..102
David A. Madore, Orders of non-abelian simple groups
R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3

Cf. A109379, A080683.

A330584 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the automorphism groups of sublattices of the Leech lattice.

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 20160, 20160, 25920, 62400, 95040, 126000, 181440, 443520, 604800, 979200, 1451520, 1814400, 3265920, 4245696, 10200960

Offset: 1

Hal M. Switkay, Dec 18 2019

Note: not every sublattice of the Leech lattice is necessarily a section of the Leech lattice. For example, every Niemeyer lattice is commensurable with the Leech lattice; thus the orders of the simple components of their automorphism groups are in this list, even when those groups are not sections of Co0.
By a theorem of Conway and Sloane, any simple group with a cover that has a crystallographic representation in <= 21 dimensions is in this list.
This is a subsequence of A330583.

			All simple groups of order less than 9828 have crystallographic representations within sublattices of the Leech lattice. The smallest nontrivial crystallographic representation of L2(27), of order 9828, is 26-dimensional.

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].
J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer, 3rd ed., 1999.

Hal M. Switkay, Table of n, a(n) for n = 1..56
J. H. Conway, N. J. A. Sloane, Low-dimensional lattices V: Integral coordinates for integral lattices, Proc. Royal Soc. A 426 (1989), 211-232.
David A. Madore, Orders of non-abelian simple groups
R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3

Cf. A109379, A080683, A330583.

A330585 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the sporadic finite simple groups.

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 12180, 14880, 20160, 20160, 25920, 29120, 32736, 58800, 62400, 95040, 102660, 126000, 175560, 178920, 181440, 265680, 372000, 443520, 604800

Offset: 1

Hal M. Switkay, Dec 18 2019

By the classification theorem for finite simple groups, there are exactly 26 sporadic finite simple groups, whose orders form A001228. The online ATLAS includes lists of the maximal subgroups of these groups, and entries for their simple subquotients.

Subsequence of A083207. - Ivan N. Ianakiev, Jan 02 2020

			This list includes the orders of all non-cyclic simple groups of order less than 9828. L2(27), of order 9828, does not appear as a subquotient of any of the sporadic finite simple groups.

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].

Hal M. Switkay, Table of n, a(n) for n = 1..82
Ivan N. Ianakiev, Subsequence of A083207, Proof
David A. Madore, Orders of non-abelian simple groups
R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3

Cf. A109379, A001228, A083207.

cp's OEIS Frontend

A001034 Orders of noncyclic simple groups (without repetition).

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

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

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A330583 The orders, with repetition, of the non-cyclic finite simple groups whose orders are 23-smooth.

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A330584 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the automorphism groups of sublattices of the Leech lattice.

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A330585 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the sporadic finite simple groups.

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