A059238 -id:A059238 - cp's OEIS frontend

A000252 Number of invertible 2 X 2 matrices mod n.

1, 6, 48, 96, 480, 288, 2016, 1536, 3888, 2880, 13200, 4608, 26208, 12096, 23040, 24576, 78336, 23328, 123120, 46080, 96768, 79200, 267168, 73728, 300000, 157248, 314928, 193536, 682080, 138240, 892800, 393216, 633600, 470016, 967680, 373248, 1822176, 738720

Offset: 1

N. J. A. Sloane

For a prime p, a(p) = (p^2 - 1)*(p^2 - p) (this is the order of GL(2,p)). More generally a(n) is multiplicative: if the canonical factorization of n is the Product_{i=1..k} (p_i)^(e_i), then a(n) = Product_{i=1..k} (((p_i)^(2*e_i) - (p_i)^(2*e_i - 2)) * ((p_i)^(2*e_i) - (p_i)^(2*e_i - 1))). - Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Apr 05 2001, Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
a(n) is the order of the automorphism group of the group C_n X C_n, where C_n is the cyclic group of order n. - Laszlo Toth, Dec 06 2011
Order of the group GL(2,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Jul 08 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.

The order of GL_2(K) for a finite field K is in sequence A059238.
Row n=2 of A316622.
Row sums of A316566.
Cf. A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A227499.

Table[n*EulerPhi[n]*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}], {n, 21}] (* Jean-François Alcover, Apr 04 2011, after Vladeta Jovovic *)

a(n)=my(f=factor(n)[,1]); n^4*prod(i=1,#f, (1-1/f[i]^2)*(1-1/f[i])) \\ Charles R Greathouse IV, Feb 06 2017

from math import prod
from sympy import factorint
def A000252(n): return prod(p**((e<<2)-3)*(p*(p*(p-1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = n^4*Product_{primes p dividing n} (1 - 1/p^2)*(1 - 1/p) = n^4*Product_{primes p dividing n} p^(-3)*(p^2 - 1)*(p - 1). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
Multiplicative with a(p^e) = (p - 1)^2*(p + 1)*p^(4e-3). - David W. Wilson, Aug 01 2001
a(n) = A000056(n)*phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
Dirichlet g.f.: zeta(s - 4)*Product_{p prime} (1 - p^(1 - s)*(p^2 + p - 1)). - Álvar Ibeas, Nov 28 2017
a(n) = A227499(n) for odd n; (3/4)*A227499(n) for even n. - Jianing Song, Jul 08 2018
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Aug 20 2021
Sum_{n>=1} 1/a(n) = (Pi^8/3240) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^5 + 2/p^6 - 1/p^8) = 1.2059016071... . - Amiram Eldar, Dec 03 2022

More terms from David W. Wilson, Jul 21 2001

A047927 a(n) = n(n-1)(n-2)^2.

Original entry on oeis.org

0, 6, 48, 180, 480, 1050, 2016, 3528, 5760, 8910, 13200, 18876, 26208, 35490, 47040, 61200, 78336, 98838, 123120, 151620, 184800, 223146, 267168, 317400, 374400, 438750, 511056, 591948, 682080, 782130, 892800, 1014816, 1148928, 1295910, 1456560, 1631700, 1822176

Offset: 2

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002417, A059238, A245334.

a047927 n = if n == 2 then 0 else a245334 n 3
-- Reinhard Zumkeller, Aug 31 2014

[n*(n-1)*(n-2)^2: n in [2..40]]; // Vincenzo Librandi, May 02 2011

a[n_] := n*(n-1)*(n-2)^2; Array[a, 50, 2] (* Amiram Eldar, Jan 15 2023 *)

a(n)=n*(n-1)*(n-2)^2 \\ Charles R Greathouse IV, Oct 07 2015

From R. J. Mathar, May 01 2014: (Start)
G.f.: -6*x^3*(1+3*x) / (x-1)^5.
a(n) = 6*A002417(n-2). (End)
a(n) = A245334(n,3), n > 2. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=3} 1/a(n) = Pi^2/12 - 5/8.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 - 2*log(2) + 9/8. (End)

Offset changed from 0 to 2 by Vincenzo Librandi, May 02 2011

A329119 Orders of the finite groups SL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 24, 60, 120, 336, 504, 720, 1320, 2184, 4080, 4896, 6840, 12144, 15600, 19656, 24360, 29760, 32736, 50616, 68880, 79464, 103776, 117600, 148824, 205320, 226920, 262080, 300696, 357840, 388944, 492960, 531360, 571704, 704880, 912576, 1030200, 1092624, 1224936, 1294920

Offset: 1

Jianing Song, Nov 05 2019

nonn

SL_2(K) means the group of 2 X 2 matrices A over K such that det(A) = 1.
In general, let R be any commutative ring with unity, GL_n(R) be the group of n X n matrices A over R such that det(A) != 0 and SL_n(R) be the group of n X n matrices A over R such that det(A) = 1, then GL_n(R)/SL_n(R) = R* is the multiplicative group of R. This is because if we define f(M) = det(M) for M in GL_n(R), then f is a surjective homomorphism from GL_n(K) to R*, and SL_n(R) is its kernel. Thus |GL_n(R)|/|SL_n(R)| = |R*|; if K is a finite field, then |GL_n(R)|/|SL_n(R)| = |K|-1.
Also a(n) is the order of PGL_2(K) when K is a finite field with q = A246655(n) elements. Note that PGL(m,q) and SL(m,q) are not isomorphic unless gcd(m,q-1) = 1. For example, PGL(2,3) = S_4 is not isomorphic to SL(2,3), PGL(2,5) = S_5 is not isomorphic to SL(2,5). - Jianing Song, Apr 04 2022

			a(4) = 120 because A246655(4) = 5, and 5*(5^2-1) = 120.

Robert Israel, Table of n, a(n) for n = 1..10000
Groupprops, Projective general linear group of degree two
Groupprops, Special linear group of degree two

Subsequence of A007531.
Cf. A246655, A000056 (order of SL_2(Z_n)).
For the order of GL_2(K) see A059238.

N:= 200:
P:= select(isprime, {2,seq(i,i=3..N,2)}):
PP:= map(proc(p) local i; seq(p^i,i=1..floor(log[p](N))) end proc, P):
map(t -> t*(t^2-1), sort(convert(PP,list))); # Robert Israel, Nov 13 2019

p = Select[Range[200], PrimePowerQ];
(p-1) p (p+1) (* Jean-François Alcover, Aug 22 2020 *)

[(p+1)*p*(p-1) | p <- [1..200], isprimepower(p)]

If the finite field K has q elements, then the order of the group SL_2(K) is q*(q^2-1).

a(n) = A059238(n)/(A246655(n)-1) = A007531(A246655(n)+1).

A352806 Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 12, 60, 60, 168, 504, 360, 660, 1092, 4080, 2448, 3420, 6072, 7800, 9828, 12180, 14880, 32736, 25308, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 262080, 150348, 178920, 194472, 246480, 265680, 285852, 352440, 456288, 515100, 546312, 612468, 647460

Offset: 1

Jianing Song, Apr 04 2022

nonn

For a communtative unital ring R, PSL_n(R), the projective special linear group of order n over R, is defined as SL_n(R)/{r*I_n: r^n = 1}. This is related to PGL_n(R), the projective general linear group of order n over R, which is defined as GL_n(R)/{r*I_n: r is a unit of R}.

Note that a(3) = a(4) = 60 refer to the same group (PSL(2,4) = PSL(2,5) = Alt(5)). Also PSL(2,9) = Alt(6).

			a(6) = 504 since A246655(6) = 8, so a(6) = 8*(8^2-1)/gcd(2,8-1) = 504.
a(7) = 360 since A246655(7) = 9, so a(7) = 9*(9^2-1)/gcd(2,9-1) = 360.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Projective special linear group

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): this sequence;
Aut(GL(2,q)): A353247;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.
A117762 is a subsequence, A335000 is a supersequence.

[(q+1)*q*(q-1)/gcd(2,q-1) | q <- [1..200], isprimepower(q)]

|PSL(2,q)| = q*(q^2-1)/2 if q is odd, q*(q^2-1) otherwise.
|PSL(2,q)| = |PGL(2,q)|/gcd(2,q-1) = |SL(2,q)|/gcd(2,q-1).
In general, |PSL(n,q)| = |PGL(n,q)|/gcd(n,q-1) = |SL(n,q)|/gcd(n,q-1).

A352807 Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 24, 120, 120, 336, 1512, 1440, 1320, 2184, 16320, 4896, 6840, 12144, 31200, 58968, 24360, 29760, 163680, 50616, 68880, 79464, 103776, 235200, 148824, 205320, 226920, 1572480, 300696, 357840, 388944, 492960, 2125440, 571704, 704880, 912576, 1030200, 1092624

Offset: 1

Jianing Song, Apr 04 2022

nonn

PGammaL_n(K) is the projective semilinear group of order n over K (see Wikipedia link). It is the semidirect product of PGL_n(K) and Aut(K), where Aut(K) is the group of field automorphisms of K. So if p is a prime, then PGammaL(n,p) is isomorphic to PGL(n,p).
We also have Aut(SL_n(K)) = Aut(PGL_n(K)) = Aut(PSL_n(K)) for arbitrary field K, and when n = 2 this is isomorphic to PGammaL_2(K). If n >= 3, this is isomorphic to the semidirect product of PGammaL_2(K) and C_2.
Examples are PGammaL(2,2) = S_3, PGammaL(2,3) = S_4, PGammaL(2,4) = PGammaL(2,5) = S_5, PGammaL(2,9) = Aut(S_6) = Aut(A_6).

			a(6) = 1512 since A246655(6) = 8 = 2^3, so a(6) = 3*A329119(6) = 3*504 = 1512.
a(7) = 1440 since A246655(7) = 9 = 3^2, so a(7) = 2*A329119(7) = 2*720 = 1440.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Projective semilinear group
Mathematics Stack Exchange, Do the groups SL, PGL, and PSL over a field K always have the same automorphism group?
Wikipedia, Semilinear map

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): A352806;
Aut(GL(2,q)): A353247;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): this sequence.

[(q+1)*q*(q-1)*isprimepower(q) | q <- [1..200], isprimepower(q)]

A353247 Orders of the finite groups Aut(GL_2(K)) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 48, 240, 480, 1344, 9072, 11520, 10560, 17472, 130560, 78336, 82080, 242880, 499200, 1415232, 584640, 476160, 4910400, 1214784, 2204160, 1907136, 4566144, 7526400, 7143552, 11497920, 7261440, 56609280, 12027840, 17176320, 18669312, 23662080, 136028160, 45736320, 56390400, 58404864, 82416000, 69927936

Offset: 1

Jianing Song, Apr 08 2022

nonn

For orders of Aut(SL_2(K)) = Aut(PGL_2(K)) = Aut(PSL_2(K)) see A352807.

See the Groupprops link for a formula for |Aut(GL(n,q))| in general.

			a(5) = 1344 since A246655(5) = 7, so a(5) = A352807(5)*eulerphi(2*(7-1)) = 336*4 = 1344.
a(6) = 9072 since A246655(6) = 8, so a(6) = A352807(6)*eulerphi(2*(8-1)) = 1512*6 = 9072.
a(7) = 11520 since A246655(7) = 9, so a(7) = A352807(7)*eulerphi(2*(9-1)) = 1440*8 = 15120.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Endomorphism structure of general linear group over a finite field

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): A352806;
Aut(GL(2,q)): this sequence;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.

[(q+1)*q*(q-1)*isprimepower(q)*eulerphi(2*(q-1)) | q <- [1..200], isprimepower(q)]

A060615 Number of conjugacy classes in the group GL_2(K) when K is a finite field with q = p^m for a prime p and m >= 1.

Original entry on oeis.org

3, 8, 15, 24, 48, 63, 80, 120, 168, 255, 288, 360, 528, 624, 728, 840, 960, 1023, 1368, 1680, 1848, 2208, 2400, 2808, 3480, 3720, 4095, 4488, 5040, 5328, 6240, 6560, 6888, 7920, 9408, 10200, 10608, 11448, 11880, 12768, 14640, 15624, 16128, 16383, 17160

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 13 2001

nonn

The number of conjugacy classes in the group GL_2(K) is q^2 - 1 so this sequence is a subsequence of A005563 restricted to q = prime power. The order of the group GL_2(K) is in A059238.

A000961, A005563, A059238. A diagonal of A060638.

with(numtheory): for n from 2 to 400 do if nops(ifactors(n)[2]) = 1 then printf(`%d,`, n^2-1) fi: od:

a(n) = A000961(n+2)^2 - 1. - Sean A. Irvine, Dec 04 2022

More terms from James Sellers, Apr 14 2001

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

A000252 Number of invertible 2 X 2 matrices mod n.

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A047927 a(n) = n*(n-1)*(n-2)^2.

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A329119 Orders of the finite groups SL_2(K) when K is a finite field with q = A246655(n) elements.

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A352806 Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

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A352807 Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.

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A353247 Orders of the finite groups Aut(GL_2(K)) when K is a finite field with q = A246655(n) elements.

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A060615 Number of conjugacy classes in the group GL_2(K) when K is a finite field with q = p^m for a prime p and m >= 1.

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A047927 a(n) = n(n-1)(n-2)^2.