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%I #21 Jul 16 2021 19:57:45
%S 1,2,4,6,8,12,16,18,24,32,36,48,54,64,72,96,108,120,128,144,162,192,
%T 200,216,240,256,288,324,384,400,432,480,486,512,576,648,720,768,800,
%U 864,960,972,1024,1152,1200,1296,1440,1458,1536,1600,1728,1920,1944
%N Numbers k such that there exists a finite group G of order k such that all entries in its character table are integers.
%C 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 fixed-point-free action (a Frobenius group). - Derek Holt
%C From _Eric M. Schmidt_, Feb 22 2013: (Start)
%C 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.
%C A necessary condition for a group G to have an integral character table is for G/G' to be an elementary Abelian 2-group. Therefore, by the Feit-Thompson theorem, the only odd term in this sequence is 1.
%C R. Gow proved (see link) that no prime greater than 5 can divide the order of a solvable group with integral character table. (End)
%H Roderick Gow, <a href="https://doi.org/10.1016/0021-8693(76)90098-3">Groups whose characters are rational-valued</a>, J. Algebra 40 (1976) 280-299.
%H Hegedus Pal, <a href="https://doi.org/10.1112/S0024611504015035">Structure of Solvable Rational Groups</a>, Proc. London Math. Soc. (2005) 90 (2): 439-471.
%o (GAP) HasIntegerCharTable := function(G) local cc, ccr, e; cc := ConjugacyClasses(G); ccr := List(cc, Representative); e := Exponent(G); return ForAll([2..e-1], 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
%Y Contains A000142 and A007694.
%K nonn,nice
%O 1,2
%A Tim Brooks (tim_brooks(AT)my-deja.com), Oct 07 2001
%E More terms from Derek Holt (mareg(AT)csv.warwick.ac.uk), Oct 07 2001
%E More terms from _Eric M. Schmidt_, Feb 22 2013
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