A note on permutations in an arbitrary field

L. CARLITZ1 The writer [l] has proved that every permutation on the numbers of the finite field GFiq) is generated by the special permutations (1) x"~2, ax + ß (a, ßEGFiq), a ^ 0). Let F denote an arbitrary field. Define the function far1 (* G F, x j¿ 0), (2) x* = \ K ' '' lO (* 0). Clearly x* defines a permutation of F. The following theorem holds. Theorem 1. Every transposition iaß), where a,ßEFis finitely generated by the special permutations (3) x*, yx + d iy,dEF,y^0). The proof (compare [l]) follows from consideration of the function