### Abstract

Let p>2 be a prime. A function f: GF(p)→GF(p) is planar if for every a∃GF(p) ^{*} , the function f(x+a-f(x) is a permutation of GF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. If P is a protective plane of prime order p admitting a collineation group of order p ^{2} , then P is the Galois plane PG(2, p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom [3] and our results are generalizations of some results of Johnson [8]. We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.

Original language | English |
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Pages (from-to) | 315-320 |

Number of pages | 6 |

Journal | Combinatorica |

Volume | 9 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1989 |

### Keywords

- AMS subject classification (1980): 05B25, 11T21, 51E15

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*9*(3), 315-320. https://doi.org/10.1007/BF02125898