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A211171 Exponent of general linear group GL(n,2). 1
1, 6, 84, 420, 26040, 78120, 9921240, 168661080, 24624517680, 270869694480, 554470264600560, 7208113439807280, 59041657185461430480, 2538791258974841510640, 383357480105201068106640, 98522872387036674503406480, 25826982813282567927671981480160 (list; graph; refs; listen; history; text; internal format)
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

StackExchange thread on the exponent of GL(n,q)

FORMULA

a(n) = 2^ceiling(Log_2(n)) * ( product from k=1 to n of the k-th cyclotomic polynomial evaluated at 2 ).

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;

CROSSREFS

Cf. A003418, A053651.

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

Sequence in context: A067249 A288321 A155191 * A054605 A119576 A098982

Adjacent sequences:  A211168 A211169 A211170 * A211172 A211173 A211174

KEYWORD

nonn

AUTHOR

Alexander Gruber, Jan 31 2013

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)