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A002884 Number of nonsingular n X n matrices over GF(2); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).
(Formerly M4302 N1798)

%I M4302 N1798

%S 1,1,6,168,20160,9999360,20158709760,163849992929280,

%T 5348063769211699200,699612310033197642547200,

%U 366440137299948128422802227200,768105432118265670534631586896281600

%N Number of nonsingular n X n matrices over GF(2); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).

%C Also number of bases for GF(2^n) over GF(2).

%C Also (apparently) number of n X n matrices over GF(2) having permanent = 1. - _Hugo Pfoertner_, Nov 14 2003

%C The previous comment is true because over GF(2) permanents and determinants are the same. - _Joerg Arndt_, Mar 07 2008

%C The number of automorphisms of (Z_2)^n (the direct product of n copies of Z_2). - _Peter Eastwood_, Apr 06 2015

%D Carter, Roger W. Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28. John Wiley & Sons, London-New York-Sydney, 1972. viii+331pp. MR0407163 (53 #10946). See page 2.

%D J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985, p. xvi.

%D H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

%D Horadam, K. J., Hadamard matrices and their applications. Princeton University Press, Princeton, NJ, 2007. xiv+263 pp. See p. 132.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe and Seiichi Manyama, <a href="/A002884/b002884.txt">Table of n, a(n) for n = 0..57</a> (first 30 terms from T. D. Noe)

%H Marcus Brinkmann, <a href="https://www.researchgate.net/publication/332564840_Extended_Affine_and_CCZ_Equivalence_Classes_up_to_Dimension_4">Extended Affine and CCZ Equivalence up to Dimension 4</a>, Ruhr University Bochum (2019).

%H Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.

%H Zong Duo Dai, Solomon W. Golomb, Guang Gong, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00035-7">Generating all linear orthomorphisms without repetition</a>, Discrete Math. 205 (1999), 47-55.

%H P. F. Duvall, Jr. and P. W. Harley, III, <a href="http://www.jstor.org/stable/2099958">A note on counting matrices</a>, SIAM J. Appl. Math., 20 (1971), 374-377.

%H N. Ilievska, D. Gligoroski, <a href="https://doi.org/10.1007/978-3-319-09879-1_31">Error-Detecting Code Using Linear Quasigroups</a>, ICT Innovations 2014, Advances in Intelligent Systems and Computing Volume 311, 2015, pp 309-318.

%H A. Meyerowitz & N. J. A. Sloane, <a href="/A002884/a002884.pdf">Correspondence 1979</a>

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%H J. Overbey, W. Traves and J. Wojdylo, <a href="http://jeff.over.bz/papers/undergrad/on-the-keyspace-of-the-hill-cipher.pdf">On the Keyspace of the Hill Cipher</a>, Cryptologia, Volume 29, 2005 - Issue 1.

%H I. Strazdins, <a href="http://dx.doi.org/10.1023/A:1005769927571">Universal affine classification of Boolean functions</a>, Acta Applic. Math. 46 (1997), 147-167.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%F Product(2^n-2^i, i=0..n-1); or 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n).

%F a(n) = A203303(n+1)/A203303(n). - _R. J. Mathar_, Jan 06 2012

%F a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)) for n > 2. - _Seiichi Manyama_, Oct 20 2016

%F a(n) ~ A048651 * 2^(n^2). - _Vaclav Kotesovec_, May 19 2020

%e PSL_2(2) is isomorphic to the symmetric group S_3 of order 6.

%p product(2^n-2^i,i=0..n-1); or 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n);

%t Table[Product[2^n-2^i,{i,0,n-1}],{n,0,13}] (* _Harvey P. Dale_, Aug 07 2011 *)

%t Table[2^(n*(n-1)/2) QPochhammer[2, 2, n] // Abs, {n, 0, 11}] (* _Jean-Fran├žois Alcover_, Jul 15 2017 *)

%o (PARI) a(n)=prod(i=2,n,2^i-1)<<binomial(n,2) \\ _Charles R Greathouse IV_, Jan 13 2012

%Y Column k=2 of A316622 and A316623.

%Y Cf. A000409, A000410, A002820, A005329, A046747, A048651, A028365.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

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Last modified July 13 07:54 EDT 2020. Contains 335685 sequences. (Running on oeis4.)