Note on permutations in a finite field
K.D. Fryer · Proceedings of the American Mathematical Society · 1955
Hence Q in standard form is a product of transpositions. If m is a square of GF(q), Q leaves 0 and two other elements fixed. Then Q is an odd permutation if q=4n+1, and even if q=4j+3. Q has the reverse character if m is a nonsquare of GF(q), for then only 0 is left fixed by Q, and Q contains an extra transposition. Hence {P, Q} contains even and odd permutations if (2) or (3) holds, but only even permutations if (4) or (5) holds. But {P, Q} contains the permutation