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A085495
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Maximal order of a finite subgroup of the group GL(n,Q).
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0
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2, 12, 48, 1152, 3840, 103680, 2903040, 696729600, 1393459200, 8360755200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000
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OFFSET
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1,1
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COMMENTS
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a(n) is also the maximal degree over Q of an algebraic number that has what the authors call conjugate dimension equal n.
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)
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LINKS
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FORMULA
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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.
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 15 2003
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EXTENSIONS
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Terms 2, 4, 6, 7, 8, 9, 10 corrected by M. F. Hasler, Dec 17 2007
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STATUS
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approved
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