Displaying 1-10 of 23 results found.
Number of n X n binary matrices of order dividing 3 (also number of solutions to X^3=I in GL(n,2)).
+10
30
1, 3, 57, 1233, 75393, 19109889, 6326835201, 6388287561729, 23576681450405889, 120906321631678693377, 1968421511613895105052673, 111055505036706392268074909697, 8965464105556083354144035638870017
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
PROG
(PARI) \\ See Morison theorem 2.6
\\ F(n, q, k) is number of solutions to X^k=I in GL(i, GF(q)) for i=1..n.
\\ q is power of prime and gcd(q, k) = 1.
B(n, q, e)={sum(m=0, n\e, x^(m*e)/prod(k=0, m-1, q^(m*e)-q^(k*e)))}
F(n, q, k)={if(gcd(q, k)<>1, error("no can do")); my(D=ffgen(q)^0); my(f=factor(D*(x^k-1))); my(p=prod(i=1, #f~, (B(n, q, poldegree(f[i, 1])) + O(x*x^n))^f[i, 2])); my(r=B(n, q, 1)); vector(n, i, polcoeff(p, i)/polcoeff(r, i))}
Number of solutions of x^10=1 in general affine group AGL(n,2).
+10
17
2, 10, 92, 23200, 21391520, 35841831040, 95709758320640, 6206883395497062400, 1502803598296957497344000, 654083813715060854940290252800, 450433384822340709737677746549555200
Number of solutions of x^2=1 in general affine group AGL(n,2).
+10
16
2, 10, 92, 1696, 59552, 4124800, 556101632, 148425895936, 78099471368192, 81705857229783040, 169694608681978560512, 702657511446831375056896, 5797142351555426979908943872, 95500953266115919784543392890880, 3140561514292519005433439594146168832
Number of n X n binary matrices of order dividing 5 (i.e., number of solutions of X^5=I in GL(n,2)).
+10
12
1, 1, 1, 1345, 666625, 223985665, 65019838465, 105072058957825, 11436238073940148225, 997931868985434228916225, 74706800043914446529756135425, 5321514758546715999509008953114625, 3721818216683598164434468712927276826625
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
PROG
(PARI) \\ See A053725 for F(n, q, k).
Number of n X n binary matrices of order dividing 12 (i.e., number of solutions of X^12=I in GL(n,2)).
+10
11
1, 6, 120, 10368, 2582208, 3143720448, 11692182896640, 219197554267521024, 12804488375721592356864, 3325324798296500862330077184, 2537067900325971750395878897090560, 8900626797123384385697033838119859781632, 65799342288255766009804607851267459830106816512
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
Number of n X n binary matrices of order dividing 6 (i.e., number of solutions of X^6=I in GL(n,2)).
+10
5
1, 6, 78, 6588, 1332288, 1335398688, 2230748717184, 13819713971871744, 219439188546028498944, 16360198814356838801178624, 3333281205541847127897252298752, 2704161270841324410691567986117967872
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
Number of n X n binary matrices of order dividing 8 (i.e., number of solutions of X^8=I in GL(n,2)).
+10
5
1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 1989505896802466922496, 164384949539438492410445824, 47902612878717208996830483841024
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
Number of cyclic subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).
+10
4
1, 5, 79, 6974, 2037136, 2890467344, 14011554132032, 325330342132674560, 27173394819858612320256, 10158190320726534408118452224, 13156630408268153048253765001412608, 80280189722884518774834501142737770774528
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
FORMULA
a(n) = Sum_{d} |{g element of A_n(2): order(g)=d}|/phi(d), where phi=Euler totient function, cf. A000010.
EXAMPLE
a(3) = 1/phi(1)+21/phi(2)+56/phi(3)+42/phi(4)+48/phi(7) = 79.
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 01 2001
Number of solutions of x^3=1 in general affine group AGL(n,2).
+10
4
1, 9, 225, 6273, 968193, 307091457, 144510377985, 338450286215169, 1535613392752345089, 11693653105154832465921, 423384155808298738368118785, 29155340360444250715547947237377
Number of n X n binary matrices of order dividing 7 (i.e., number of solutions of X^7=I in GL(n,2)).
+10
3
1, 1, 49, 5761, 476161, 457113601, 3439085027329, 18696142934507521, 144017748317668638721, 30063679011292374997401601, 10371304522603231166854078660609, 3639433320096084212920229480292679681, 18767347744724322162378748108305552459694081
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
PROG
(PARI) \\ See A053725 for F(n, q, k).
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