A329119 -id:A329119 - cp's OEIS frontend

A059238 Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements.

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

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 21 2001

nonn

From Jianing Song, Nov 06 2019: (Start)
GL_2(K) means the group of invertible 2 X 2 matrices A over K.
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. (End)

			a(4) = 480 because A246655(4) = 5, and (5^2-1)*(5^2-5) = 480.

Jianing Song, Table of n, a(n) for n = 1..10000
R. A. Wilson, The classical groups, chapter 3.3.1 in The finite Simple Groups, Graduate Texts in Mathematics 251 (2009).

Subsequence of A047927.
Cf. A246655, A000252 (order of GL_2(Z_n)).
For the order of SL_2(K) see A329119.

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

nn=30;a=Take[Union[Sort[Flatten[Table[Table[Prime[m]^k,{m,1,nn}],{k,1,nn}]]]],nn];Table[(q^2-1)(q^2-q),{q,a}]  (* Geoffrey Critzer, Apr 20 2013 *)

[(p+1)*p*(p-1)^2 | p <- [1..200], isprimepower(p)] \\ Jianing Song, Nov 05 2019

If the finite field K has p^m elements, then the order of the group GL_2(K) is (p^(2m)-1)*(p^(2m)-p^m) = (p^m+1)*(p^m)*(p^m-1)^2.
a(n) = A047927(A246655(n)+1). - Jianing Song, Nov 05 2019
a(n) = (A246655(n)-1)*A329119(n). - Jianing Song, Nov 06 2019

More terms from James Sellers, Jan 22 2001

Offset corrected by Jianing Song, Nov 05 2019

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)]

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A059238 Orders of the finite groups GL_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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