A334994 -id:A334994 - cp's OEIS frontend

A335000 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (with repetitions).

6, 12, 60, 60, 168, 168, 360, 504, 660, 1092, 2448, 3420, 4080, 5616, 6072, 7800, 9828, 12180, 14880, 20160, 20160, 25308, 32736, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 262080, 265680, 285852, 352440, 372000, 456288, 515100, 546312

Offset: 1

Michel Marcus, May 19 2020

nonn

60 is the order of PSL(2,4) and of PSL(2,5).
168 is the order of PSL(2,7) and of PSL(3,2).
20160 is the order of PSL(4,2) and of PSL(3,4).
Other repetitions > 20160 for PSL(m,q) groups are not known.
See A334884 and A334994 for variations of this sequence.

			a(5) = #PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168, and,
a(6) = #PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168.

Wikipedia, Projective linear group.
Index entries for sequences related to groups.

Cf. A002884 \ {1} (PSL(n,2)), A117762 (PSL(2, prime(n))).

Cf. A334884 (another case with repetitions), A334994 (without repetitions).

#PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1). - Bernard Schott, May 19 2020

A334884 Order of the non-isomorphic groups PSL(m,q) [or PSL_m(q)] in increasing order as q runs through the prime powers.

Original entry on oeis.org

6, 12, 60, 168, 360, 504, 660, 1092, 2448, 3420, 4080, 5616, 6072, 7800, 9828, 12180, 14880, 20160, 20160, 25308, 32736, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 262080, 265680, 285852, 352440, 372000, 456288

Offset: 1

Bernard Schott, May 14 2020

nonn

The projective special linear group PSL(m,q) is the quotient group of SL(m,q) with its center.
Theorem: The group PSL(m,q) is simple except for PSL(2,2) and PSL(2,3).
Exceptional isomorphisms (let "==" denote "isomorphic to"):
a(1) = 6 for PSL(2,2) == GL(2,2) == SL(2,2) == S_3 (see example).
a(2) = 12 for PSL(2,3) == A_4.
a(3) = 60 for PSL(2,4) and for PSL(2,5) with PSL(2,4) == PSL(2,5) == A_5 that is the smallest nonabelian simple group.
a(4) = 168 for PSL(2,7) and for PSL(3,2) with PSL(2,7) == PSL(3,2); PSL(2, 7) is the second smallest nonabelian simple group (see example).
a(5) = 360 for PSL(2,9) == A_6.
a(18) = a(19) = 20160 for PSL(4,2) == A_8 and for PSL(3,4) non-isomorphic to A_8 (see comment in A137863).
Array for order of PSL(m,q):
m\q|  2         3            4 =2^2            5               7
----------------------------------------------------------------------
2 |   6         12             60              60             168
3 |  168       5616           20160          372000         1876896
4 | 20160    6065280        987033600      7254000000    2317591180800
5 | 9999360 237783237120  258492255436800 56653740000000000  #PSL(5,7)
              with #PSL(5,7) = 187035198320488089600

			a(1) = #PSL(2,2) = (2^2-1)*2 = 6 and the 6 elements of PSL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 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(4) = #PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168, and also,
a(4) = #PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168.

Wikipedia, Projective linear group

Subsequence: A117762 (PSL(2,prime(n))).
Cf. A137863.
Cf. A334994 and A335000 for other versions of this sequence.

#PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1).

cp's OEIS Frontend

A335000 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (with repetitions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula