cp's OEIS Frontend

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

%I A327568 #14 Aug 13 2021 02:35:55
%S A327568 1,6,24,12,120,24,336,24,72,120,1320,24,2184,336,120,48,4896,72,6840,
%T A327568 120,336,1320,12144,24,600,2184,216,336,24360,120,29760,96,1320,4896,
%U A327568 1680,72,50616,6840,2184,120,68880,336,79464,1320,360,12144
%N A327568 Exponent of the group GL(2, Z_n).
%C A327568 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.
%H A327568 Kenneth G. Hawes, <a href="/A327568/b327568.txt">Table of n, a(n) for n = 1..200</a>
%H A327568 The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Exponent_of_a_group">Exponent of a group</a>
%F A327568 If gcd(m, n) = 1 then a(m*n) = lcm(a(m), a(n)).
%F A327568 Conjecture: a(p^e) = (p^2-1)*p^e for primes p. If this is true, then 24 divides a(n) for n > 2.
%e A327568 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.
%o A327568 (PARI) MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++; N=N*M); k}
%o A327568 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
%Y A327568 Cf. A000252, A316565, A327569.
%K A327568 nonn
%O A327568 1,2
%A A327568 _Jianing Song_, Sep 17 2019