cp's OEIS Frontend

A064527 Numbers k such that there exists a finite group G of order k such that all entries in its character table are integers.

%I A064527 #24 Oct 13 2024 00:17:24
%S A064527 1,2,4,6,8,12,16,18,24,32,36,48,54,64,72,96,108,120,128,144,162,192,
%T A064527 200,216,240,256,288,324,384,400,432,480,486,512,576,648,720,768,800,
%U A064527 864,960,972,1024,1152,1200,1296,1440,1458,1536,1600,1728,1920,1944
%N A064527 Numbers k such that there exists a finite group G of order k such that all entries in its character table are integers.
%C A064527 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 A064527 From _Eric M. Schmidt_, Feb 22 2013: (Start)
%C A064527 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 A064527 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 A064527 R. Gow proved (see link) that no prime greater than 5 can divide the order of a solvable group with integral character table. (End)
%C A064527 From _Jianing Song_, Oct 12 2024: (Start)
%C A064527 A finite group whose all characters are rational valued is usually called a Q-group of a rational group, although different authors many define these terms differently.
%C A064527 The unique rational group of order 200 is SmallGroup(200,44) (see Tim Dokchitser's link below). (End)
%H A064527 Tim Dokchitser, <a href="https://people.maths.bris.ac.uk/~matyd/GroupNames/RG.html">Rational groups</a>
%H A064527 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 A064527 Groupprops, <a href="https://groupprops.subwiki.org/wiki/Finite_rational_group">Finite rational group</a>
%H A064527 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 A064527 (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 A064527 Contains A000142 and A007694.
%K A064527 nonn,nice
%O A064527 1,2
%A A064527 Tim Brooks (tim_brooks(AT)my-deja.com), Oct 07 2001
%E A064527 More terms from Derek Holt (mareg(AT)csv.warwick.ac.uk), Oct 07 2001
%E A064527 More terms from _Eric M. Schmidt_, Feb 22 2013