A316563 -id:A316563 - cp's OEIS frontend

A316564 Triangle read by rows: T(n,k) is the number of elements of the group SL(2, Z(n)) with order k, 1 <= k <= A316563(n).

1, 1, 3, 2, 1, 1, 8, 6, 0, 8, 1, 7, 8, 24, 0, 8, 1, 1, 20, 30, 24, 20, 0, 0, 0, 24, 1, 7, 26, 24, 0, 74, 0, 0, 0, 0, 0, 12, 1, 1, 56, 42, 0, 56, 48, 84, 0, 0, 0, 0, 0, 48, 1, 15, 32, 144, 0, 96, 0, 96, 1, 1, 98, 54, 0, 98, 0, 0, 144, 0, 0, 108, 0, 0, 0, 0, 0, 144

Offset: 1

Andrew Howroyd, Jul 06 2018

For coprime p,q the group SL(p*q, Z(n)) is isomorphic to the direct product of the two groups SL(p, Z(n)) and SL(q, Z(n)).

			Triangle begins:
  1;
  1,  3,  2;
  1,  1,  8, 6, 0, 8;
  1,  7,  8, 24, 0, 8;
  1,  1, 20, 30, 24, 20, 0, 0, 0, 24;
  1,  7, 26, 24, 0, 74, 0, 0, 0, 0, 0, 12;
  1,  1, 56, 42, 0, 56, 48, 84, 0, 0, 0, 0, 0, 48;
  1, 15, 32, 144, 0, 96, 0, 96;
  1,  1, 98, 54, 0, 98, 0, 0, 144, 0, 0, 108, 0, 0, 0, 0, 0, 144;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..3478 (first 60 rows)

Column 2 is A316553.
Row sums are A000056.
Cf. A316537, A316566, A316586.

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
row(n)={my(L=List()); 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, my(t=MatOrder(M)); while(#L

T(p*q,k) = Sum_{i>0, j>0, k=lcm(i, j)} T(p, i)*T(q, j) for gcd(p, q)=1.

T(n,k) = Sum_{d|k} mu(d/k) A316586(n,d).

A316537 Number of cyclic subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

Original entry on oeis.org

1, 5, 13, 28, 49, 73, 116, 176, 202, 265, 378, 464, 550, 636, 842, 936, 1041, 1183, 1486, 1712, 2082, 2055, 2120, 3088, 2114, 3023, 2503, 4200, 4238, 4862, 4902, 4648, 6564, 5749, 7434, 7688, 6331, 8190, 9880, 11344, 10172, 12066, 9378, 13224, 14168, 11612

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

			Case n=2: generators of the 5 cyclic groups are:
  [ 1 0 ]   [0 1]   [1 0]   [1 1]   [0 1]
  [ 0 1 ]   [1 0]   [1 1]   [0 1]   [1 1]

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000056, A062314, A316536, A316553, A316560, A316563, A316564.

Concatenation([1], List([2..10], n->Sum( Filtered( ConjugacyClassesSubgroups( SL(2, Integers mod n)), x->IsCyclic( Representative(x))), Size)));

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
a(n)={sum(a=0, n-1, sum(b=0, n-1, sum(c=0, n-1, sum(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, 1/eulerphi(MatOrder(M)))))))}

a(n) = Sum_{k=1..A316563(n)} 1/phi(A316564(n, k)).

A316565 Maximum order of an element of the general linear group GL(2, Z(n)).

Original entry on oeis.org

1, 3, 8, 6, 24, 24, 48, 12, 24, 60, 120, 24, 168, 48, 60, 24, 288, 24, 360, 60, 168, 330, 528, 24, 120, 168, 72, 84, 840, 120, 960, 48, 440, 816, 420, 36, 1368, 360, 312, 60, 1680, 168, 1848, 330, 180, 1518, 2208, 48, 336, 300, 816, 168, 2808, 72, 1320, 168

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

Row lengths of A316566.

Cf. A000252, A316563.

Concatenation([1], List([2..10], n->Maximum(List(GL(2, Integers mod n), Order))));

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
a(n)={my(m=0); 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(gcd(lift(matdet(M)), n)==1, m=max(m, MatOrder(M))))))); m}

Conjecture: a(p) = (p-1)*(p+1) for prime p.
From Robert Israel, Dec 19 2019: (Start)
The conjecture is true.  In fact for T in GL(2,Z(p)), the order of T divides p*(p-1) if the characteristic polynomial of T splits over Z(p) and p^2-1 if it doesn't; moreover, if T is the companion matrix of the minimal polynomial of a primitive element of GF(p^2), the order is p^2-1.
a(p^k) <= (p^2-1) p^(k-1).
If m and n are coprime, a(m*n) <= a(m)*a(n). (End)

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.

cp's OEIS Frontend

A316564 Triangle read by rows: T(n,k) is the number of elements of the group SL(2, Z(n)) with order k, 1 <= k <= A316563(n).

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A316537 Number of cyclic subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

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GAP

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A316565 Maximum order of an element of the general linear group GL(2, Z(n)).

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Keywords

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Programs

GAP

PARI

Formula

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula