A085495 Maximal order of a finite subgroup of the group GL(n,Q).

2, 12, 48, 1152, 3840, 103680, 2903040, 696729600, 1393459200, 8360755200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000

Offset

1,1

Comments

a(n) is also the maximal degree over Q of an algebraic number that has what the authors call conjugate dimension equal n.

From Jianing Song, May 06 2021: (Start)

a(n) is also the maximal order of a finite subgroup of the group GL(n,Z). This is because every finite subgroup of GL(n,Q) is conjugate to a subgroup of GL(n,Z). See the Math Overflow link below.

Conjecture: a(n)/2 is the maximal order of a finite subgroup of the group SL(n,Q) (or equivalently, SL(n,Z), since a finite subgroup of SL(n,Q) must be conjugate to a subgroup of SL(n,Z)). This is obviously true for odd n: given a finite subgroup G of SL(n,Q), a subgroup of GL(n,Q) of order 2*|G| can be attained by adjoining -I_n where I_n is the identity matrix. The conjecture is also correct for n = 2, since the finite subgroup of the maximal order of SL(n,Q) is isomorphic to C_6. (End)

Links

Table of n, a(n) for n=1..17.

N. Berry, A. Dubickas, N. D. Elkies, B. Poonen and C. Smyth, The conjugate dimension of algebraic numbers, arXiv:math/0308069 [math.NT], 2003-2004; Quart. J. Math., 55 (2004), 237-252.

Math Overflow, Reference for a linear algebra result

Formula

For all n other than the seven exceptional values 2, 4, 6, 7, 8, 9, 10, a(n) = A000165(n) = 2^n * n! and the relevant group is the group of n X n signed permutation matrices.

Crossrefs

Cf. A000165.

Sequence in context: A052588 A139239 A060238 * A119978 A139234 A039784

Adjacent sequences: A085492 A085493 A085494 * A085496 A085497 A085498

Keyword

nonn,easy

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 15 2003

Extensions

More terms from N. J. A. Sloane, Dec 20 2006

Terms 2, 4, 6, 7, 8, 9, 10 corrected by M. F. Hasler, Dec 17 2007

Status

approved