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

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

Author

Andrew Howroyd, Jul 06 2018

Keywords

• nonn

Crossrefs

Row lengths of A316566.

Cf. A000252, A316563.

Programs

GAP

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

PARI

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}

Formula

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)