

A064527


Numbers k such that there exists a finite group G of order k 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 m! for all m because the symmetric groups have integral character tables. By taking direct products, we get all numbers of the form m! * 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
A group of order k has integral character table iff g^m is conjugate to g for all group elements g and all m coprime to k.
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 link) that no prime greater than 5 can divide the order of a solvable group with integral character table. (End)


LINKS



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



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


STATUS

approved



