A000056 -id:A000056 - cp's OEIS frontend

A115226 Order of the group of invertible 3 X 3 symmetric matrices over Z(n).

1, 28, 468, 1792, 12400, 13104, 100548, 114688, 341172, 347200, 1609300, 838656, 4453488, 2815344, 5803200, 7340032, 22713088, 9552816, 44563284, 22220800, 47056464, 45060400, 141587908, 53673984, 193750000, 124697664, 248714388, 180182016, 574288624, 162489600

Offset: 1

T. D. Noe, Jan 16 2006

Note that A115225 gives the number of 3 x 3 symmetric matrices having nonzero determinant. However, for composite n, a nonzero determinant is not sufficient for the matrix to be invertible; the determinant must also be relatively prime to n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000056 (order of the group SL(2, Z_n)), A064767 (order of the group GL(3, Z_n)), A115225.

Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, Modulus->n]], cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}]; cnt, {n, 2, 20}]
f[p_, e_] := p^(6*e - 4)*(p^3 - 1)*(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(6*f[i,2] - 4)*(f[i,1]^3 - 1)*(f[i,1] - 1));} \\ Amiram Eldar, Nov 05 2022

For prime p, a(p) = (p^3-1)*(p-1)*p^2.
In general, a(n) = A115224(n) * phi(n) = A064767(n)/A000056(n).
Multiplicative with a(p^e) = p^(6*e - 4)*(p^3 - 1)*(p - 1). - Amiram Eldar, Sep 10 2020
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^4/((p-1)^3 * (p^2+p+1)^2 * (p^3+1))) = 1.03859354030263389220782701124174403591851545785245128014455467710993780757... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.08230753362... . - Amiram Eldar, Nov 05 2022

More terms from Amiram Eldar, Sep 10 2020

A300915 Order of the group PSL(2,Z_n).

Original entry on oeis.org

1, 6, 12, 24, 60, 72, 168, 96, 324, 360, 660, 288, 1092, 1008, 720, 768, 2448, 1944, 3420, 1440, 2016, 3960, 6072, 1152, 7500, 6552, 8748, 4032, 12180, 4320, 14880, 6144, 7920, 14688, 10080, 7776, 25308, 20520, 13104, 5760

Offset: 1

Geoffrey Critzer, Mar 16 2018

The projective special linear group PSL(2,Z_n) is the quotient group of SL(2,Z_n) with its center. The center of SL(2,Z_n) is the group of scalar matrices whose diagonal entry is x in Z_n such that x^2 = 1. The elements of PSL(2,Z_n) are equivalence classes of 2 X 2 matrices with entries in Z_n where two matrices are equivalent if one is a scalar multiple of the other.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Cf. A000056, A060594.

n := 2; nn = 40; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; Prepend[ Table[Product[ FactorInteger[m][[All, 1]][[j]]^(n^2 (FactorInteger[m][[All, 2]][[j]] - 1)) \[Gamma][n,FactorInteger[m][[All, 1]][[j]]], {j, 1, PrimeNu[m]}], {m, 2, nn}]/Table[EulerPhi[m], {m, 2, nn}]/ Table[Count[Mod[Select[Range[m], GCD[#, m] == 1 &]^n, m], 1], {m, 2, nn}], 1]
f[p_, e_] := (p^2-1)*p^(3*e-2)/2; f[2, e_] := Switch[e, 1, 6, 2, 24, , 3*2^(3*e-4)]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Dec 01 2022 *)

a(n) = {my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); (p^2-1)*p^(3*e-2)/if(p==2, 2^min(2, e-1), 2))} \\ Andrew Howroyd, Aug 01 2018

a(n) = A000056(n)/A060594(n).

Multiplicative with a(2) = 6, a(2^2) = 24, a(2^e) = 3*2^(3*e-4) for e > 2, and a(p^e) = (p^2-1)*p^(3*e-2)/2 for p > 2. - Amiram Eldar, Dec 01 2022

Keyword:mult added by Andrew Howroyd, Aug 01 2018

A316498 Number of normal subgroups of the special linear group SL(2, Z_n).

Original entry on oeis.org

1, 3, 4, 7, 3, 13, 3, 15, 6, 10, 3, 32, 3, 10, 13, 26, 3, 20, 3, 25, 13, 10, 3, 75, 5, 10, 8, 25, 3, 48, 3, 37, 13, 10, 10, 50, 3, 10, 13, 60, 3, 48, 3, 25, 20, 10, 3, 135, 5, 17, 13, 25, 3, 27, 10, 60, 13, 10, 3, 127, 3, 10, 20, 48, 10, 48, 3, 25, 13, 38

Offset: 1

Andrew Howroyd, Jul 04 2018

nonn

Cf. A000056, A066541.

Concatenation([1], List([2..20], n->Size(NormalSubgroups(SL(2, Integers mod n)))));

A327569 Exponent of the group SL(2, Z_n).

Original entry on oeis.org

1, 6, 12, 12, 60, 12, 168, 24, 36, 60, 660, 12, 1092, 168, 60, 48, 2448, 36, 3420, 60, 168, 660, 6072, 24, 300, 1092, 108, 168, 12180, 60, 14880, 96, 660, 2448, 840, 36, 25308, 3420, 1092, 120, 34440, 168, 39732, 660, 180, 6072, 51888, 48, 1176, 300, 2448, 1092, 74412, 108, 660, 168

Offset: 1

Jianing Song, Sep 17 2019

nonn

The exponent of a finite group G is the least positive integer k such that x^k = e for all x in G, where e is the identity of the group. That is to say, the exponent of a finite group G is the LCM of the orders of elements in G. Of course, the exponent divides the order of the group.

			SL(2, Z_2) is isomorphic to S_3, which has 1 identity element, 3 elements with order 2 and 2 elements with order 3, so a(2) = lcm(1, 2, 3) = 6.

The Group Properties Wiki, Exponent of a group

Cf. A000056, A316563, A327568.

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++; N=N*M); k}
a(n)={my(m=1); for(a=0, n-1, for(b=0, n-1, for(c=0, n-1, for(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, m=lcm(m, MatOrder(M))))))); m} \\ Following Andrew Howroyd's program for A316563

If gcd(m, n) = 1 then a(m*n) = lcm(a(m), a(n)).

Conjecture: a(p^e) = (p^2-1)*p^e/2 for primes p > 2 and 3*2^e for p = 2. If this is true, then 12 divides a(n) for n > 2.

A194894 The number of the ordered triples (A,B,C) satisfying the system of the modular relations {AB - BA = C, BC - CB = A, CA - AC = B}, where A,B,C are distinct 2 X 2 matrices over Z(n).

Original entry on oeis.org

0, 0, 24, 0, 120, 24, 336, 0, 648, 120, 1320, 24, 2184, 336, 3024, 0, 4896, 648, 6840, 120, 8424, 1320, 12144, 24, 15000, 2184, 17496, 336, 24360, 3024, 29760, 0, 33024, 4896, 40776, 648, 50616, 6840, 54624, 120, 68880

Offset: 1

Erdos Pal, Sep 04 2011

If (A,B,C) is a triple and X is chosen from among A,B,C, then trace(X)=0 mod n, X*X = -det(X)*IdentityMatrix mod n, A*B + B*A = B*C + C*B = C*A + A*C = 0 mod n, det(A) = det(B) = det(C) mod n, A*A = B*B = C*C mod n, A = 2*B*C, B = 2*C*A, C = 2*A*B mod n.
For a given value of n, consider the family of triples (A,B,C) for which d = det(A) = det(B) = det(C) mod n. Let b(n,d) denote the number of elements of the set {A: (A,B,C) is a triple and det(A) = d}. Let b(n) = Sum{ b(n,d) for all such d }, for example, d(15) = 6 + 30 + 180. Detailed results of searching for trios (N(d) = number of triples in the family):
. .n b(n,d) ...d ......N
. .1 .....0 .... ......0
. .2 .....0 .... ......0
. .3 .....6 ...1 .....24
. .4 .....0 .... ......0
. .5 ....30 ...4 ....120
. .6 .....6 ...4 .....24
. .7 ....42 ...2 ....336
. .8 .....0 .... ......0
. .9 ....54 ...7 ....648
. 10 ....30 ...4 ....120
. 11 ...110 ...3 ...1320
. 12 .....6 ...4 .....24
. 13 ...182 ..10 ...2184
. 14 ....42 ...2 ....336
. 15......6 ..10 .....24
. 15.....30 ...9 ....120
. 15....180 ...4 ...2880
. 16 .....0 .... ......0
. 17 ...306 ..13 ...4896
. 18 ....54 ..16 ....648
. 19 ...342 ...5 ...6840
. 20 ....30 ...4 ....120
. 21......6 ...7 .....24
. 21....252 ..16 ...8064
. 21.....42 ...9 ....336
. 22 ...110 ..14 ...1320
. 23 ...506 ...6 ..12144
. 24 .....6 ..16 .....24
. 25 ...750 ..19 ..15000
. 26 ...182 .... ...2184
. 27 ...486 ...7 ..17496
. 28 ....42 ..16 ....336
. 29 ...870 ..22 ..24360
. 30......6 ..10 .....24
. 30.....30 ..24 ....120
. 30....180 ...4 ...2880
. 31 ...930 ...8 ..29760
. 32 .....0 .... ......0
. 33......6 ..22 .....24
. 33....660 ..25 ..31680
. 33....110 ...3 ...1320
. 34 ...306 ..30 ...4896
. 35...1260 ...9 ..40320
. 35.....42 ..30 ....336
. 35.....30 ..14 ....120
. 36 ....54 ..16 ....648
. 37 ..1406 ..28 ..50616
. 38 ...342 ..24 ...6840
. 39......6 ..13 .....24
. 39....182 ..36 ...2184
. 39...1092 ..10 ..52416
. 40 ....30 ..24 ....120
. 41 ..1722 ..31 ..68880
Remarks for the cases n<=41 (conjectures for n>41):
b(n) is similar to a(n), i.e., b(2^e)=0 for e>=0, b(m*2^e)=b(m) for m>=0 and e>=0, b(m*n) = b(m) + b(n) + b(m)*b(n) for gcd(m,n)=1;
b(p) = (p-1)*p for primes of the form p = 4*k + 1;
b(p) = p*(p+1) for primes of the form p = 4*k - 1;
b(p^e) = b(p)*(p^(2*(e-1))) for odd primes p and e>=1;
if n=p^e (p is odd prime, e>=1) then d is a constant for all trios (there is only one family), moreover 4*d=1 (mod n).

			The matrices A=[0,1;2,0], B=[1,1;1,2], C=[2,1;1,1] of row order form satisfy the system of the (mod 3)-relations {A*B - B*A = C, A#B, B*C - C*B = A, B#C, C*A - A*C = B, C#A}, so we have a trio (+A,+B,+C). All the solutions of the system can be represented by the trios
(+A,+B,+C), (+B,+C,+A), (+C,+A,+B),
(+A,-C,+B), (-C,+B,+A), (+B,+A,-C),
(+A,+C,-B), (+C,-B,+A), (-B,+A,+C),
(+A,-B,-C), (-B,-C,+A), (-C,+A,-B),
(-A,+B,-C), (+B,-C,-A), (-C,-A,+B),
(-A,-C,-B), (-C,-B,-A), (-B,-A,-C),
(-A,+C,+B), (+C,+B,-A), (+B,-A,+C),
(-A,-B,+C), (-B,+C,-A), (+C,-A,-B), so a(3)=24.

Cf. A000056, A181107.

a(2^e) = 0 for e>=0; a( m*(2^e) ) = a(m) for m>=1,e>=0.
a(p^e) = (p^2-1)*p^(3*e-2) for odd prime p,e>=1.
a(m*n) = a(m) + a(n) + a(m)*a(n) for gcd(m,n)=1

cp's OEIS Frontend

A115226 Order of the group of invertible 3 X 3 symmetric matrices over Z(n).

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A300915 Order of the group PSL(2,Z_n).

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A316498 Number of normal subgroups of the special linear group SL(2, Z_n).

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A327569 Exponent of the group SL(2, Z_n).

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A194894 The number of the ordered triples (A,B,C) satisfying the system of the modular relations {A*B - B*A = C, B*C - C*B = A, C*A - A*C = B}, where A,B,C are distinct 2 X 2 matrices over Z(n).

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A194894 The number of the ordered triples (A,B,C) satisfying the system of the modular relations {AB - BA = C, BC - CB = A, CA - AC = B}, where A,B,C are distinct 2 X 2 matrices over Z(n).