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%I #33 May 24 2020 10:39:00
%S 6,12,60,168,360,504,660,1092,2448,3420,4080,5616,6072,7800,9828,
%T 12180,14880,20160,20160,25308,32736,34440,39732,51888,58800,74412,
%U 102660,113460,150348,178920,194472,246480,262080,265680,285852,352440,372000,456288
%N Order of the non-isomorphic groups PSL(m,q) [or PSL_m(q)] in increasing order as q runs through the prime powers.
%C The projective special linear group PSL(m,q) is the quotient group of SL(m,q) with its center.
%C Theorem: The group PSL(m,q) is simple except for PSL(2,2) and PSL(2,3).
%C Exceptional isomorphisms (let "==" denote "isomorphic to"):
%C a(1) = 6 for PSL(2,2) == GL(2,2) == SL(2,2) == S_3 (see example).
%C a(2) = 12 for PSL(2,3) == A_4.
%C 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.
%C 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).
%C a(5) = 360 for PSL(2,9) == A_6.
%C 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).
%C Array for order of PSL(m,q):
%C m\q| 2 3 4 =2^2 5 7
%C ----------------------------------------------------------------------
%C 2 | 6 12 60 60 168
%C 3 | 168 5616 20160 372000 1876896
%C 4 | 20160 6065280 987033600 7254000000 2317591180800
%C 5 | 9999360 237783237120 258492255436800 56653740000000000 #PSL(5,7)
%C with #PSL(5,7) = 187035198320488089600
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>
%F #PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1).
%e 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:
%e (1 0) (1 1) (1 0) (0 1) (0 1) (1 1)
%e (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
%e a(4) = #PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168, and also,
%e a(4) = #PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168.
%Y Subsequence: A117762 (PSL(2,prime(n)).
%Y Cf. A137863.
%Y Cf. A334994 and A335000 for other versions of this sequence.
%K nonn
%O 1,1
%A _Bernard Schott_, May 14 2020