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A211171 Exponent of general linear group GL(n,2). 1

%I

%S 1,6,84,420,26040,78120,9921240,168661080,24624517680,270869694480,

%T 554470264600560,7208113439807280,59041657185461430480,

%U 2538791258974841510640,383357480105201068106640,98522872387036674503406480,25826982813282567927671981480160

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

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

%H Alexander Gruber, <a href="/A211171/b211171.txt">Table of n, a(n) for n = 1..100</a>

%H <a href="http://math.stackexchange.com/a/294524/12952">StackExchange thread</a> on the exponent of GL(n,q)

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

%e n = 2: GL(2,2) is isomorphic to S3 which has exponent 6 (see: A003418).

%e n = 3: The set of element orders of GL(3,2) is {1,2,3,4,7} so the exponent is 84.

%e 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).

%p with(numtheory):

%p a:= proc(n) local t; t:= 2^ilog2(n);

%p `if`(t<n, 2, 1)*t*mul(cyclotomic(k, 2), k=1..n)

%p end:

%p seq(a(n), n=1..20); # _Alois P. Heinz_, Feb 04 2013

%t f[q_, n_] := With[{p = Sort[Divisors[q]][[2]]},

%t p^Ceiling[Log[p, n]] Product[Cyclotomic[k, q], {k, n}]]; f[2,#]&/@Range[100]

%o (MAGMA)

%o for n in [1..18] do

%o Exponent(GL(n,2));

%o end for;

%Y Cf. A003418, A053651.

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

%K nonn

%O 1,2

%A _Alexander Gruber_, Jan 31 2013

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Last modified November 19 20:42 EST 2019. Contains 329323 sequences. (Running on oeis4.)