cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Jianing Song, Apr 04 2022

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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.

Programs

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

Formula

For q = p^r, |PGammaL(2,q)| = r*q*(q^2-1) = r*|PGL(2,q)|. In general, |PGammaL(n,q)| = r*|PGL(n,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

Views

Author

Jianing Song, Apr 08 2022

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Programs

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

Formula

For q = p^r, |Aut(GL(2,q))| = r*q*(q^2-1)*eulerphi(2*(q-1)) = |PGammaL(2,q)|*eulerphi(2*(q-1)) (see A352807). In general, we have |Aut(GL(n,q))|/|Aut(SL(n,q))| = eulerphi(n*(q-1))/eulerphi(n).

A364004 Orders of simple groups PSL(2,K) with exactly 4 prime divisors.

Original entry on oeis.org

660, 1092, 4080, 3420, 6072, 7800, 9828, 14880, 32736, 25308, 51888, 58800, 74412, 194472, 265680, 456288, 612468, 1024128, 2097024, 2165292, 3594432, 7174332, 8487168, 28090752, 57750408, 96049728, 321367392
Offset: 1

Views

Author

Lixin Zheng, Jul 03 2023

Keywords

Comments

Sequence is conjectured to be infinite, see Bugeaud et al.
All entries are divisible by 6 by order formula for PSL(2,q).

Examples

			660 has prime divisors 2,3,5,11.

Links

  • Y. Bugeaud, Z. Cao, and M. Mignotte, On Simple K4-Groups, Journal of Algebra, 241 (2001), 658-668.

Crossrefs

Subsequence of A352806. Elements generated from A364003.

Formula

Terms are q*(q^2-1)/gcd(2, q-1) for q in A364003.
a(n) = A033931(A364003(n)-1).
Showing 1-3 of 3 results.

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