Puzzle 7: Solution

Q     [A problem I contributed to the 1988 Qualifying Exam at Harvard]
Prove that a finite group of 8k elements (k=1,2,3,...) must have at least five conjugacy classes.

A     The number of conjugacy classes of a finite group G equals the number of irreducible representations of G. The order of G is the sum of the squares of the dimensions of these representations. At least one of these dimensions is 1 (the trivial representation). When |G|=8k, it is not possible to write |G|-1=8k-1 as a sum of fewer than four squares. Therefore the number of representations, which is also the number of conjugacy classes, is at least 4+1=5, QED.

Remarks: Equality holds for at least three groups G, namely the dihedral and quaternion groups of order 8, and the fourth symmetric group, of order 4!=24. Michael Larsen observed that for each m there is an effective upper bound on the size of a finite group with at most m conjugacy classes; when m=5, this reduces the problem, and the determination of the cases of equality, to a finite (though possibly impractical) computation. The upper bound is a consequence of the “class equation”: |G| is the sum of the sizes of its conjugacy classes, each of which is a factor of |G|, and at least one of which equals 1.