A053651 -id:A053651 - cp's OEIS frontend

A062766 Number of nonisomorphic cyclic subgroups of general affine group AGL(n,2).

2, 4, 6, 11, 16, 23, 32, 46, 62, 83, 114, 158, 212, 276, 351, 442, 562, 726, 933, 1191, 1498, 1876

Offset: 1

Vladeta Jovovic, Jul 17 2001

			a(5)=16 because the set of orders of elements of AGL(5,2) is {1,2,3,4,5,6,7,8,10,12,14,15,21,28,30,31}.

Vladeta Jovovic, Cycle index of general affine group AGL(n,2)
Index entries for sequences related to groups

Cf. A062710 (labeled case), A053651.

a(13)-a(22) from Sean A. Irvine, Jan 11 2022

A053658 Number of nonisomorphic cyclic subgroups of general linear group GL(n,3).

Original entry on oeis.org

2, 6, 8, 18, 26, 42, 62, 94, 130, 188, 264, 372, 506, 678

Offset: 1

Vladeta Jovovic, Mar 22 2000

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Cf. A053651, A053660.

a(11)-a(15) from Sean A. Irvine, Jan 10 2022

A053660 Number of nonisomorphic cyclic subgroups of general linear group GL(n,4).

Original entry on oeis.org

2, 6, 12, 18, 36, 56, 92, 138, 220, 338, 520

Offset: 1

Vladeta Jovovic, Mar 22 2000

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Cf. A053651, A053658.

a(8)-a(11) from Sean A. Irvine, Jan 10 2022

A062250 Number of cyclic subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).

Original entry on oeis.org

1, 5, 79, 6974, 2037136, 2890467344, 14011554132032, 325330342132674560, 27173394819858612320256, 10158190320726534408118452224, 13156630408268153048253765001412608, 80280189722884518774834501142737770774528

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 01 2001

nonn

			a(3) = 1/phi(1)+21/phi(2)+56/phi(3)+42/phi(4)+48/phi(7) = 79.

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

V. Jovovic, Cycle index of general linear group GL(n,2)

Cf. A053651 (unlabeled case), A053658, A053660, A053718, A053722, A053725, A053771-A053777, A058502, A062552, A062240.

a(n) = Sum_{d} |{g element of A_n(2): order(g)=d}|/phi(d), where phi=Euler totient function, cf. A000010.

More terms from Vladeta Jovovic, Jul 04 2001

A316560 Number of cyclic subgroups of the group GL(2, Z(n)), counting conjugates as distinct.

Original entry on oeis.org

1, 5, 28, 62, 176, 148, 610, 696, 1252, 920, 2296, 1972, 4874, 3523, 6040, 6320, 8136, 7348, 14984, 13568, 22124, 11920, 17396, 23952, 29846, 28172, 38044, 47656, 47282, 32908, 75036, 53520, 71768, 42312, 145852, 99892, 123524, 88456, 187036, 179200, 152290

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

Cf. A053651, A066947, A316537, A316559, A316565, A316566.

Concatenation([1], List([2..7], n->Sum( Filtered( ConjugacyClassesSubgroups( GL(2, Integers mod n)), x->IsCyclic( Representative(x))), Size)));

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
a(n)={sum(a=0, n-1, sum(b=0, n-1, sum(c=0, n-1, sum(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(gcd(lift(matdet(M)), n)==1, 1/eulerphi(MatOrder(M)))))))}

a(n) = Sum_{k=1..A316565(n)} 1/phi(A316566(n,k)).

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

Original entry on oeis.org

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

Offset: 1

Alexander Gruber, Jan 31 2013

nonn

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

			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).

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)

Cf. A003418, A053651.
Cf. A006951 (number of conjugacy classes in GL(n,2)).
Cf. A034268, A062383.

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

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

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

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

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

cp's OEIS Frontend

A062766 Number of nonisomorphic cyclic subgroups of general affine group AGL(n,2).

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A053658 Number of nonisomorphic cyclic subgroups of general linear group GL(n,3).

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A053660 Number of nonisomorphic cyclic subgroups of general linear group GL(n,4).

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A062250 Number of cyclic subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).

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A316560 Number of cyclic subgroups of the group GL(2, Z(n)), counting conjugates as distinct.

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

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