A211171 Exponent of general linear group GL(n,2).

1, 6, 84, 420, 26040, 78120, 9921240, 168661080, 24624517680, 270869694480, 554470264600560, 7208113439807280, 59041657185461430480, 2538791258974841510640, 383357480105201068106640, 98522872387036674503406480, 25826982813282567927671981480160

Offset: 1

Author

Alexander Gruber, Jan 31 2013

Keywords

• nonn

Comments

a(n) is the smallest integer for which x^a(n) = 1 for any x in GL(n,2).

Examples

			n = 2: GL(2,2) is isomorphic to S3 which has exponent 6 (see: A003418).
n = 3: The set of element orders of GL(3,2) is {1,2,3,4,7} so the exponent is 84.
n = 5: The set of element orders of GL(5,2) is {1,2,3,4,5, 6,7,8,12,14, 15,21,31} so the exponent is 26040 (see: A053651).
		

Links

• Alexander Gruber, Table of n, a(n) for n = 1..100
• Gert Almkvist, Powers of a matrix with coefficients in a Boolean ring, Proc. Amer. Math. Soc. 53 (1975), 27-31. See u_n.
• Eugene Karolinsky and Dmytro Seliutin, Carmichael numbers for GL(m), arXiv:2001.10315 [math.NT], 2020; where a(n) is noted as K2(n), see page 1.
• MathStackExchange, Exponent of GL(n,q)

Crossrefs

Cf. A003418, A053651.

Cf. A006951 (number of conjugacy classes in GL(n,2)).

Cf. A034268, A062383.

Programs

Magma

for n in [1..18] do
Exponent(GL(n,2));
end for;

Maple

with(numtheory):
a:= proc(n) local t; t:= 2^ilog2(n);
      `if`(tAlois P. Heinz, Feb 04 2013

Mathematica

f[q_, n_] := With[{p = Sort[Divisors[q]][[2]]},
  p^Ceiling[Log[p, n]] Product[Cyclotomic[k, q], {k, n}]]; f[2,#]&/@Range[100]

PARI

a(n) = 2^ceil(log(n)/log(2))*prod(k=1, n, polcyclo(k, 2)); \\ Michel Marcus, Jan 29 2020

Formula

a(n) = 2^ceiling(log_2(n)) * Product_{k=1..n} (k-th cyclotomic polynomial evaluated at 2).

a(n) = A034268(n)*A062383(n+1). - Michel Marcus, Jul 29 2022