## Permutation polynomials and primitive permutation groups.(English)Zbl 0766.11047

In 1966 L. Carlitz conjectured that for every even positive integer $$n$$ there exists a constant $$c_ n$$ such that for any odd $$q>c_ n$$ there is no permutation polynomial of degree $$n$$ over the finite field $$F_ q$$ of order $$q$$. This conjecture was known to hold for $$n$$ a power of 2 and for all even $$n\leq 16$$. In this paper the conjecture is shown for all even $$n<1000$$ and for every $$n$$ that is twice a prime. The proof exploits the connections between permutation polynomials, exceptional polynomials, and primitive permutation groups in a skillful manner. The limitation $$n<1000$$ stems from the list of all primitive groups of degree $$<1000$$ compiled by J. D. Dixon and B. Mortimer [Math. Proc. Camb. Philos. Soc. 103, 213-238 (1988; Zbl 0646.20003)].
We note that the case where $$n$$ is twice a prime was settled independently by D. Wan [Proc. Am. Math. Soc. 110, 303-309 (1990; Zbl 0711.11050)] who used a different method. The Carlitz conjecture was recently proved in full generality by M. Fried, R. Guralnick and J. Saxl.

### MSC:

 11T06 Polynomials over finite fields 20B15 Primitive groups

### Citations:

Zbl 0646.20003; Zbl 0711.11050
Full Text:

### References:

 [1] M. Aschbacher andL. Scott, Maximal subgroups of finite groups. J. Algebra92, 44-80 (1985). · Zbl 0549.20011 [2] S. D. Cohen, The distribution of polynomials over finite fields. Acta Arith.17, 255-271 (1970). · Zbl 0209.36001 [3] S. D. Cohen, The factorable core of polynomials over finite fields. J. Austral. Math. Soc. Ser. A,49, 309-318 (1990). · Zbl 0728.11065 [4] J. D. Dixon andB. Mortimer, The primitive permutation groups of degree less than 1000. Math Proc. Cambridge Philos. Soc.103, 213-238 (1988). · Zbl 0646.20003 [5] M. Fried, On a conjecture of Schur. Michigan Math. J.17, 41-55 (1970). · Zbl 0169.37702 [6] M. Fried, Arithmetical properties of function fields II. The generalized Schur problem. Acta Arith.25, 225-258 (1974). · Zbl 0229.12020 [7] M. Fried, Exposition on an arithmetic-group theoretic connection via Riemann’s existence theorem. Proc. Sympos. Pure Math.37, 571-602 (1980). · Zbl 0451.14011 [8] D. R. Hayes, A geometric approach to permutation polynomials over a finite field. Duke Math J.34, 293-305 (1967). · Zbl 0163.05202 [9] R. Lidl andG. L. Mullen, When does a polynomial over a finite field permute the elements of the field? Amer. Math. Monthly95, 243-246 (1988). · Zbl 0653.12010 [10] R.Lidl and H.Niederreiter, Finite fields. Reading, Mass. 1983. · Zbl 0554.12010 [11] D.Passman, Permutation groups. New York 1968. · Zbl 0179.04405 [12] W. R.Scott, Group Theory. Englewood Cliffs 1964. [13] T.Tsuzuku, Finite groups and finite geometries. Cambridge 1976. · Zbl 0368.50009 [14] D. Wan, On a conjecture of Carlitz. J. Austral. Math. Soc. Ser. A43, 375-384 (1987). · Zbl 0635.12011 [15] D.Wan, Permutation polynomials and resolution of singularities. Proc. Amer. Math. Soc., to appear. · Zbl 0711.11050 [16] H.Wielandt, Finite permutation groups. New York-London 1964. · Zbl 0138.02501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.