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A085495
Maximal order of a finite subgroup of the group GL(n,Q).
0
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
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.
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
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