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, 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

Offset: 1

Author

Tim Brooks (tim_brooks(AT)my-deja.com), Oct 07 2001

Keywords

• nonn
• nice

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 fixed-point-free action (a Frobenius group). - Derek Holt

From Eric M. Schmidt, Feb 22 2013: (Start)

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 2-group. Therefore, by the Feit-Thompson 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)

From Jianing Song, Oct 12 2024: (Start)

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.

The unique rational group of order 200 is SmallGroup(200,44) (see Tim Dokchitser's link below). (End)

Links

• Tim Dokchitser, Rational groups
• Roderick Gow, Groups whose characters are rational-valued, J. Algebra 40 (1976) 280-299.
• Groupprops, Finite rational group
• Hegedus Pal, Structure of Solvable Rational Groups, Proc. London Math. Soc. (2005) 90 (2): 439-471.

Crossrefs

Contains A000142 and A007694.

Programs

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

Extensions

More terms from Derek Holt (mareg(AT)csv.warwick.ac.uk), Oct 07 2001

More terms from Eric M. Schmidt, Feb 22 2013