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A327569 Exponent of the group SL(2, Z_n). 1


%S 1,6,12,12,60,12,168,24,36,60,660,12,1092,168,60,48,2448,36,3420,60,

%T 168,660,6072,24,300,1092,108,168,12180,60,14880,96,660,2448,840,36,

%U 25308,3420,1092,120,34440,168,39732,660,180,6072,51888,48,1176,300,2448,1092,74412,108,660,168

%N Exponent of the group SL(2, Z_n).

%C 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 The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Exponent_of_a_group">Exponent of a group</a>

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

%F 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.

%e 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.

%o MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++; N=N*M); k}

%o 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

%Y Cf. A000056, A316563, A327568.

%K nonn

%O 1,2

%A _Jianing Song_, Sep 17 2019

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Last modified April 9 20:55 EDT 2020. Contains 333363 sequences. (Running on oeis4.)