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A327569 Exponent of the group SL(2, Z_n). 1
1, 6, 12, 12, 60, 12, 168, 24, 36, 60, 660, 12, 1092, 168, 60, 48, 2448, 36, 3420, 60, 168, 660, 6072, 24, 300, 1092, 108, 168, 12180, 60, 14880, 96, 660, 2448, 840, 36, 25308, 3420, 1092, 120, 34440, 168, 39732, 660, 180, 6072, 51888, 48, 1176, 300, 2448, 1092, 74412, 108, 660, 168 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..56.

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/2 for primes p > 2 and 3*2^e for p = 2. If this is true, then 12 divides a(n) for n > 2.

EXAMPLE

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.

PROG

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(matdet(M)==1, m=lcm(m, MatOrder(M))))))); m} \\ Following Andrew Howroyd's program for A316563

CROSSREFS

Cf. A000056, A316563, A327568.

Sequence in context: A181797 A185152 A028588 * A221433 A004749 A107687

Adjacent sequences:  A327565 A327566 A327568 * A327570 A327571 A327572

KEYWORD

nonn

AUTHOR

Jianing Song, Sep 17 2019

STATUS

approved

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Last modified February 17 12:10 EST 2020. Contains 331996 sequences. (Running on oeis4.)