A327568 Exponent of the group GL(2, Z_n).

1, 6, 24, 12, 120, 24, 336, 24, 72, 120, 1320, 24, 2184, 336, 120, 48, 4896, 72, 6840, 120, 336, 1320, 12144, 24, 600, 2184, 216, 336, 24360, 120, 29760, 96, 1320, 4896, 1680, 72, 50616, 6840, 2184, 120, 68880, 336, 79464, 1320, 360, 12144

Offset: 1

Author

Jianing Song, Sep 17 2019

Keywords

• nonn

Comments

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.

Examples

			GL(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.
		

Links

• Kenneth G. Hawes, Table of n, a(n) for n = 1..200
• The Group Properties Wiki, Exponent of a group

Crossrefs

Cf. A000252, A316565, A327569.

Programs

PARI

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(gcd(lift(matdet(M)), n)==1, m=lcm(m, MatOrder(M))))))); m} \\ Following Andrew Howroyd's program for A316565

Formula

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

Conjecture: a(p^e) = (p^2-1)*p^e for primes p. If this is true, then 24 divides a(n) for n > 2.