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

%I A334884 #36 Dec 21 2024 01:01:46
%S A334884 6,12,60,168,360,504,660,1092,2448,3420,4080,5616,6072,7800,9828,
%T A334884 12180,14880,20160,20160,25308,32736,34440,39732,51888,58800,74412,
%U A334884 102660,113460,150348,178920,194472,246480,262080,265680,285852,352440,372000,456288
%N A334884 Order of the non-isomorphic groups PSL(m,q) [or PSL_m(q)] in increasing order as q runs through the prime powers.
%C A334884 The projective special linear group PSL(m,q) is the quotient group of SL(m,q) with its center.
%C A334884 Theorem: The group PSL(m,q) is simple except for PSL(2,2) and PSL(2,3).
%C A334884 Exceptional isomorphisms (let "==" denote "isomorphic to"):
%C A334884 a(1) = 6 for PSL(2,2) == GL(2,2) == SL(2,2) == S_3 (see example).
%C A334884 a(2) = 12 for PSL(2,3) == A_4.
%C A334884 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 A334884 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 A334884 a(5) = 360 for PSL(2,9) == A_6.
%C A334884 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 A334884 Array for order of PSL(m,q):
%C A334884 m\q|  2         3            4 =2^2            5               7
%C A334884 ----------------------------------------------------------------------
%C A334884 2 |   6         12             60              60             168
%C A334884 3 |  168       5616           20160          372000         1876896
%C A334884 4 | 20160    6065280        987033600      7254000000    2317591180800
%C A334884 5 | 9999360 237783237120  258492255436800 56653740000000000  #PSL(5,7)
%C A334884               with #PSL(5,7) = 187035198320488089600
%H A334884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>
%F A334884 #PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1).
%e A334884 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 A334884    (1 0)   (1 1)   (1 0)   (0 1)   (0 1)   (1 1)
%e A334884    (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
%e A334884 a(4) = #PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168, and also,
%e A334884 a(4) = #PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168.
%Y A334884 Subsequence: A117762 (PSL(2,prime(n))).
%Y A334884 Cf. A137863.
%Y A334884 Cf. A334994 and A335000 for other versions of this sequence.
%K A334884 nonn
%O A334884 1,1
%A A334884 _Bernard Schott_, May 14 2020