A031963 -id:A031963 - cp's OEIS frontend

A145719 Erroneous version of A031963.

1, 21, 56, 0, 42, 0, 0, 48

Offset: 1

dead

H. Ferguson and C. Ferguson, Eightfold way: the sculpture, pp. 133-173 in S. Levy, ed., The Eightfold Way, Cambridge, 1999.

A145437 a(n) = number of elements of order n in simple group Alt(12) of order 239500800.

1, 63855, 776600, 3825360, 4809024, 25530120, 570240, 29937600, 26611200, 25945920, 43545600, 21621600, 0, 8553600, 6652800, 0, 0, 0, 0, 11975040, 11404800, 0, 0, 0, 0, 0, 0, 0, 0, 3991680, 0, 0, 0, 0, 13685760

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

t1:=[0 : n in [1..240]];
G:=Alt(12);
t2:=Classes(G);
for c in t2 do
t1[c[1]] := t1[c[1]] + c[2];
end for;
t1;

A145822 a(n) = number of elements of order n in simple group Alt(11) of order 19958400.

Original entry on oeis.org

1, 18315, 142010, 457380, 809424, 2044350, 237600, 2494800, 2217600, 498960, 3628800, 2910600, 0, 712800, 887040, 0, 0, 0, 0, 997920, 1900800

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A086859 a(n) = number of elements of order n in simple group L_2(8) of order 504.

Original entry on oeis.org

1, 63, 56, 0, 0, 0, 216, 0, 168

Offset: 1

N. J. A. Sloane, Apr 06 2009

Cf. A031963.

A102578 a(n) = number of elements of order n in simple group Alt(6) = L_2(9) of order 360.

Original entry on oeis.org

1, 45, 80, 90, 144

Offset: 1

N. J. A. Sloane, Apr 06 2009

Cf. A031963. A row of A057740.

A145752 a(n) = number of elements of order n in simple group Alt(7) of order 2520.

Original entry on oeis.org

1, 105, 350, 630, 504, 210, 720

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A145753 a(n) = number of elements of order n in simple group Alt(8) of order 20160.

Original entry on oeis.org

1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A145754 a(n) = number of elements of order n in simple group Alt(9) of order 181440.

Original entry on oeis.org

1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A145770 a(n) = number of elements of order n in simple group Alt(10) of order 1814400.

Original entry on oeis.org

1, 5355, 31040, 94500, 78624, 201600, 86400, 226800, 403200, 90720, 0, 302400, 0, 0, 120960, 0, 0, 0, 0, 0, 172800

Offset: 1

N. J. A. Sloane, Apr 06 2009

A row of A057740. Cf. A031963.

A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

Original entry on oeis.org

6, 48, 168, 180, 480, 2016, 3528, 5760, 11232, 13200, 20160, 26208, 61200, 78336, 123120, 181440, 267168, 374400, 511056, 682080, 892800, 1014816, 1488000, 1822176, 2755200, 3337488, 4773696, 5644800, 7738848, 9999360, 11908560, 13615200, 16511040, 19845936, 24261120, 25048800, 28003968

Offset: 1

Bernard Schott, Jun 04 2020

nonn

GL(m,q) is the general linear group, the group of invertible m X m matrices over the finite field F_q with q = p^k elements.
By definition, all fields must contain at least two distinct elements, so q >= 2. As GL(1,q) is isomorphic to F_q*, the multiplicative group [whose order is p^k-1 (A181062)] of finite field F_q, data begins with m >= 2.
Some isomorphisms (let "==" denote "isomorphic to"):
a(1) = 6 for GL(2,2) == PSL(2,2) == S_3.
a(2) = 48 for GL(2,3) that has 55 subgroups.
a(3) = 168 for GL(3,2) == PSL(2,7) [A031963].
a(11) = 20160 for GL(4,2) == PSL(4,2) == Alt(8).
Array for order of GL(m,q) begins:
=============================================================
m\q  |      2         3       4=2^2             5         7
-------------------------------------------------------------
  2  |      6        48         180           480      2016
  3  |    168     11232      181440       1488000  33784128
  4  |  20160  24261120  2961100800  116064000000   #GL(4,7)
  5  |9999360  #GL(5,3)      ...           ...         ...

			a(1) = #GL(2,2) = (2^2-1)*(2^2-2) = 3*2 = 6 and the 6 elements of GL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 invertible matrices with entries in F_2:
  (1 0)   (1 1)   (1 0)   (0 1)   (0 1)   (1 1)
  (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.

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].
Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.

Wikipedia, General linear group

Cf. A059238 [GL(2,q)].
Cf. A002884 [GL(m,2)], A053290 [GL(m,3)], A053291 [GL(m,4)], A053292 [GL(m,5)], A053293 [GL(m,7)], A052496 [GL(m,8)], A052497 [GL(m,9)], A052498 [GL(m,11)].
Cf. A316622 [GL(n,Z_k)].

#GL(m,q) = Product_{k=0..m-1}(q^m-q^k).

cp's OEIS Frontend

A145719 Erroneous version of A031963.

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A145437 a(n) = number of elements of order n in simple group Alt(12) of order 239500800.

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A145822 a(n) = number of elements of order n in simple group Alt(11) of order 19958400.

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A086859 a(n) = number of elements of order n in simple group L_2(8) of order 504.

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A102578 a(n) = number of elements of order n in simple group Alt(6) = L_2(9) of order 360.

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A145752 a(n) = number of elements of order n in simple group Alt(7) of order 2520.

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A145753 a(n) = number of elements of order n in simple group Alt(8) of order 20160.

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A145754 a(n) = number of elements of order n in simple group Alt(9) of order 181440.

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A145770 a(n) = number of elements of order n in simple group Alt(10) of order 1814400.

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A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

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