OFFSET
1,2
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.
LINKS
Kenneth G. Hawes, Table of n, a(n) for n = 1..200
The Group Properties Wiki, Exponent of a group
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.
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 17 2019
STATUS
approved