group of the finite field of order p. (For p=2 a similar configuration occurs in the group SL2(2q), with PU a Borel subgroup of upper triangular matrices and Q the subgroup of order 3 generated by .) To eliminate this final case, Thompson used some fearsomely complicated manipulations with generators and relations, which were later simplified by , whose argument is reproduced in . The proof examines the set of elements a in the finite field of order pq such that a and 2–a both have norm 1. One first checks that this set has at least one element other than 1. Then a rather difficult argument using generators and relations in the group G shows that the set is closed under taking inverses. If a is in the set and not equal to 1 then the polynomial N((1–a)x+1)–1 has degree q and has at least p distinct roots given by the elements x in Fp, using the fact that x→1/(2–x) maps the set to itself, so p≤q, contradicting the assumption p>q. Use of oddness The fact that the order of the group G is odd is used in several places in the proof, as follows . The Hall–Higman theorem is sharper for groups of odd order. For groups of odd order, all non-principal characters occur in complex conjugate pairs. Several results about p-groups only hold for odd primes p. If a group of odd order has no elementary abelian subgroups of rank 3, then its derived group is nilpotent. (This fails for the symmetric group S4 of even order.) Several arguments involving character theory fail for small primes, especially for the prime 2. References Theorems about finite groups