

A064527


Numbers n such that there exists a finite group G of order n such that all entries in its character table are integers.


1



1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 120, 128, 144, 162, 192, 200, 216, 240, 256, 288, 324, 384, 400, 432, 480, 486, 512, 576, 648, 720, 768, 800, 864, 960, 972, 1024, 1152, 1200, 1296, 1440, 1458, 1536, 1600, 1728, 1920, 1944
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OFFSET

1,2


COMMENTS

The list contains all numbers of the form 2^w*3^u for w> 0, u>=0. But it also contains 120, 200, 240 and 400. It contains n! for all n because the symmetric groups have integral character tables. By taking direct products, we get all numbers of the form n! * 2^w * 3^u, w > 0, u >= 0. The 200 comes from a semidirect product of an elementary group of order 25 with a quaternion group of order 8, with fixedpointfree action (a Frobenius group).  Derek Holt
From Eric M. Schmidt, Feb 22 2013: (Start)
A group of order n has integral character table iff g^m is conjugate to g for all group elements g and all m coprime to n.
A necessary condition for a group G to have an integral character table is for G/G' to be an elementary Abelian 2group. Therefore, by the FeitThompson theorem, the only odd term in this sequence is 1.
R. Gow proved (see reference) that no prime greater than 5 can divide the order of a solvable group with integral character table. (End)


REFERENCES

Roderick Gow, Groups whose characters are rationalvalued, J. Algebra 40 (1976) 280299.
Hegedus Pal, Structure of Solvable Rational Groups, Proc. London Math. Soc. (2005) 90 (2): 439471.


LINKS

Table of n, a(n) for n=1..53.


PROG

(GAP) HasIntegerCharTable := function(G) local cc, ccr, e; cc := ConjugacyClasses(G); ccr := List(cc, Representative); e := Exponent(G); return ForAll([2..e1], m>(not (IsPrimeInt(m) and GcdInt(m, e)=1)) or ForAll([1..Length(cc)], j>ccr[j]^m in cc[j])); end; A064527 := function(max) local res, i, j; res := [1]; for i in [2, 4..max(max mod 2)] do if ForAny(res, j>i/j in res) then Add(res, i); continue; fi; for j in [1..NumberSmallGroups(i)] do if HasIntegerCharTable(SmallGroup(i, j)) then Add(res, i); continue; fi; od; od; return res; end; # Eric M. Schmidt, Feb 22 2013


CROSSREFS

Contains A000142 and A007694.
Sequence in context: A067946 A227270 A145853 * A007694 A219653 A050622
Adjacent sequences: A064524 A064525 A064526 * A064528 A064529 A064530


KEYWORD

nonn,nice


AUTHOR

Tim Brooks (tim_brooks(AT)mydeja.com), Oct 07 2001


EXTENSIONS

More terms from Derek Holt (mareg(AT)csv.warwick.ac.uk), Oct 07, 2001
More terms from Eric M. Schmidt, Feb 22 2013


STATUS

approved



