Automorphism groups of power functions

Abstract Let 𝐹 be a finite field, and for any integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> k\geq 0 , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>p</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> p_{k} be the power function on 𝐹 defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>p</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> p_{k}(x)=x^{k} . We determine the group of CCZ automorphisms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>p</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> p_{k} , i.e. the group of invertible affine transformations which preserve the graph of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>p</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> p_{k} .