A052496 -id:A052496 - cp's OEIS frontend

A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 2, 48, 168, 1, 1, 4, 96, 11232, 20160, 1, 1, 2, 480, 86016, 24261120, 9999360, 1, 1, 6, 288, 1488000, 1321205760, 475566474240, 20158709760, 1, 1, 4, 2016, 1886976, 116064000000, 335522845163520, 84129611558952960, 163849992929280, 1

Offset: 0

Andrew Howroyd, Jul 08 2018

All rows are multiplicative.
Equivalently, the number of invertible n X n matrices mod k.
Also, for k prime (but not higher prime powers) the number of nonsingular n X n matrices over GF(k).
For k >= 2, n! divides T(n,k) since the subgroup of GL(n,k) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). Note that a permutation matrix is an orthogonal matrix, hence having determinant +-1. - Jianing Song, Oct 29 2022

			Array begins:
=================================================================
n\k| 1       2         3          4             5           6
---+-------------------------------------------------------------
0  | 1       1         1          1            1            1 ...
1  | 1       1         2          2            4            2 ...
2  | 1       6        48         96          480          288 ...
3  | 1     168     11232      86016      1488000      1886976 ...
4  | 1   20160  24261120 1321205760 116064000000 489104179200 ...
5  | 1 9999360  ...
...

R. P. Brent and B. D. McKay, Determinants and ranks of random matrices over Zm, Discrete Mathematics 66 (1987) pp. 35-49.
J. M. Lockhart and W. P. Wardlaw, Determinants of Matrices over the Integers Modulo m, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 207-214.
The Group Properties Wiki, Order formulas for linear groups

Rows n=2..4 are A000252, A064767, A305186.
Columns k=2..7 are A002884, A053290, A065128, A053292, A065498, A053293.
Cf. A053291 (GF(4)), A052496 (GF(8)), A052497 (GF(9)).
Cf. A316623.

T:=function(n,k) if k=1 or n=0 then return 1; else return Order(GL(n, Integers mod k)); fi; end;
for n in [0..5] do Print(List([1..6], k->T(n,k)), "\n"); od;

T[, 1] = T[0, ] = 1; T[n_, k_] := T[n, k] = Module[{f = FactorInteger[k], p, e}, If[Length[f] == 1, {p, e} = f[[1]]; (p^e)^(n^2)* Product[(1 - 1/p^j), {j, 1, n}], Times @@ (T[n, Power @@ #]& /@ f)]];
Table[T[n - k + 1, k], {n, 0, 8}, {k, n + 1, 1, -1}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)

T(n,k)={my(f=factor(k)); k^(n^2) * prod(i=1, #f~, my(p=f[i,1]); prod(j=1, n, (1 - p^(-j))))}

T(n,p^e) = (p^e)^(n^2) * Product_{j=1..n} (1 - 1/p^j) for prime p.

A052497 Number of nonsingular n X n matrices over GF(9).

Original entry on oeis.org

1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, Table of n, a(n) for n = 0..30
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Cf. A027877, A053764.

Cf. A002884, A003792, A053290, A053291, A053292, A053293, A052496, A052498, A132037.

[1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019

[product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).
a(n) = A053764(n)*A027877(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 9^(n^2), where c = A132037. - Amiram Eldar, Jul 06 2025

A003791 Order of universal Chevalley group A_n (8).

Original entry on oeis.org

1, 504, 16482816, 34558531338240, 4638226007491010887680, 39841906041871272087686291128320, 21903309038581548352789123727634573903790080

Offset: 0

N. J. A. Sloane

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], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003790, A003792, A052496, A132036.

[&*[(8^n - 8^k): k in [0..n-1]]/7: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[8, #] & /@ Range[0, 6] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A052496(n)/7. - Ralf Stephan, Mar 30 2004
a(n) = A(8,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 8^(n*(n+2)), where c = (8/7) * A132036 = 0.982178279315... . - Amiram Eldar, Jul 07 2025

A052498 Number of nonsingular n X n matrices over GF(11).

Original entry on oeis.org

1, 10, 13200, 2124276000, 41393302251840000, 97602635428252959312000000, 27847155251069188894843979022720000000, 961359275427083998992553051820498439890246400000000

Offset: 0

Vladeta Jovovic, Mar 16 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..30

Cf. A027879, A110195.

Cf. A002884, A053290, A053291, A053292, A053293, A052496, A052497, A132267.

[1] cat [&*[(11^n - 11^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[11^n - 11^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)

{a(n) = prod(j=0,n-1, 11^n - 11^j)}; \\ G. C. Greubel, May 14 2019

[product(11^n - 11^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (11^n - 1)*(11^n - 11)*...*(11^n - 11^(n-1)).
a(n) = A110195(n)*A027879(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 11^(n^2), where c = A132267. - Amiram Eldar, Jul 06 2025

A220790 Product(6^n - 6^k, k=0..n-1).

Original entry on oeis.org

1, 5, 1050, 8127000, 2273284440000, 22906523331216000000, 8310241106635054164480000000, 108537128570336598656772717772800000000, 51032497739317419104816901041614046625792000000000

Offset: 0

Vincenzo Librandi, Jan 28 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..35

Cf. A027873, A109354.

Sequences given by product(m^n-m^k, k=0..n-1): A002884 (m=2), A053290 (m=3), A053291 (m=4), A053292 (m=5), A053293 (m=7), A052496 (m=8), A052497 (m=9), A052498 (m=11).

[1] cat [&*[(6^n - 6^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

/* By the second formula: */
m:=9;
A109354 := [6^(n*(n-1) div 2): n in [0..m-1]];
A027873 := [1] cat [&*[6^i-1: i in [1..n]]: n in [1..m]];
[A109354[i]*A027873[i]: i in [1..m]]; // Bruno Berselli, Jan 30 2013

Table[Product[6^n - 6^k, {k, 0, n-1}], {n, 0, 60}]

a(n) = (6^n - 1)*(6^n - 6)*...*(6^n - 6^(n-1)) for n>0, a(0)=1.

a(n) = A109354(n)*A027873(n). - Bruno Berselli, Jan 30 2013

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

A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

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A052497 Number of nonsingular n X n matrices over GF(9).

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Magma

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A003791 Order of universal Chevalley group A_n (8).

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Magma

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A052498 Number of nonsingular n X n matrices over GF(11).

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A220790 Product(6^n - 6^k, k=0..n-1).

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Magma

Magma

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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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