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

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

Author

Bernard Schott, May 14 2020

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Wikipedia, Projective linear group

Crossrefs

Subsequence: A117762 (PSL(2,prime(n))).

Cf. A137863.

Cf. A334994 and A335000 for other versions of this sequence.

Formula

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