A strengthening of McConnel’s theorem on permutations over finite fields
Canadian mathematical bulletin, Tome 68 (2025) no. 1, pp. 213-218
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Let p be a prime, $q=p^n$, and $D \subset \mathbb {F}_q^*$. A celebrated result of McConnel states that if D is a proper subgroup of $\mathbb {F}_q^*$, and $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on D to obtain the same conclusion on f. In particular, we show that McConnel’s theorem extends if D has small doubling.
Yip, Chi Hoi. A strengthening of McConnel’s theorem on permutations over finite fields. Canadian mathematical bulletin, Tome 68 (2025) no. 1, pp. 213-218. doi: 10.4153/S0008439524000742
@article{10_4153_S0008439524000742,
author = {Yip, Chi Hoi},
title = {A strengthening of {McConnel{\textquoteright}s} theorem on permutations over finite fields},
journal = {Canadian mathematical bulletin},
pages = {213--218},
year = {2025},
volume = {68},
number = {1},
doi = {10.4153/S0008439524000742},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439524000742/}
}
TY - JOUR AU - Yip, Chi Hoi TI - A strengthening of McConnel’s theorem on permutations over finite fields JO - Canadian mathematical bulletin PY - 2025 SP - 213 EP - 218 VL - 68 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008439524000742/ DO - 10.4153/S0008439524000742 ID - 10_4153_S0008439524000742 ER -
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