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

1,2

Comments

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

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)

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

Example

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).

Maple

with(numtheory):
a:= proc(n) local t; t:= 2^ilog2(n);
      `if`(t<n, 2, 1)*t*mul(cyclotomic(k, 2), k=1..n)
    end:
seq(a(n), n=1..20);  # Alois 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]

Prog

(Magma)
for n in [1..18] do
Exponent(GL(n, 2));
end for;
(PARI) a(n) = 2^ceil(log(n)/log(2))*prod(k=1, n, polcyclo(k, 2)); \\ Michel Marcus, Jan 29 2020

Crossrefs

Cf. A003418, A053651.

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

Cf. A034268, A062383.

Sequence in context: A350947 A288321 A155191 * A338405 A054605 A119576

Adjacent sequences: A211168 A211169 A211170 * A211172 A211173 A211174

Keyword

nonn

Author

Alexander Gruber, Jan 31 2013

Status

approved