Skew-morphisms of elementary abelian 𝑝-groups

A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>σ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>⁢</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>σ</m:mi> <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:mo>⁢</m:mo> <m:msup> <m:mi>σ</m:mi> <m:mrow> <m:mi>π</m:mi> <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:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> σ(xy)=σ(x)σπ(x)(y) for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> x,y∈ G . It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:mi>σ</m:mi> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:math> σ forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> Zₚⁿ . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo lspace="0.222em" rspace="0.222em">⊲</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> Zₚⁿ X if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> p=2 and either <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo lspace="0.222em" rspace="0.222em">⊲</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> Zₚⁿ X or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>X</m:mi> </m:msub> <m:mo>≅</m:mo> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:math> (Zₚⁿ)Xₚⁿ⁻¹ if 𝑝 is an odd prime. As an application, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n≤ 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>AGL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> AGL(n,p) and enumerate the number of skew-morphisms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> Zₚⁿ .