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Skew-morphisms of elementary abelian 𝑝-groups

  • Shaofei Du , Wenjuan Luo EMAIL logo , Hao Yu and Junyang Zhang
Published/Copyright: May 7, 2024
Journal of Group Theory
From the journal Journal of Group Theory Volume 27 Issue 6

Abstract

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 σ ⁢ ( x ⁢ y ) = σ ⁢ ( x ) ⁢ σ π ⁢ ( x ) ⁢ ( y ) for all 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 ⟨ 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 Z p n . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that Z p n ⊲ X if p = 2 and either Z p n ⊲ X or ( Z p n ) X ≅ Z p n − 1 if 𝑝 is an odd prime. As an application, for n ≤ 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group AGL ⁢ ( n , p ) and enumerate the number of skew-morphisms of Z p n .

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12071312

Award Identifier / Grant number: 11971248

Funding statement: This work was supported by the National Natural Science Foundation of China (12071312 and 11971248).

Acknowledgements

The authors would like to thank the reviewers for their valuable comments and suggestions.

  1. Communicated by: Andrea Lucchini

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Received: 2022-05-24
Revised: 2024-04-07
Published Online: 2024-05-07
Published in Print: 2024-11-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Open mappings of locally compact groups
  3. Groups of profinite type and profinite rigidity
  4. On conjugate separability of nilpotent subgroups
  5. Groups whose proper subgroups of infinite rank have a permutability transitive relation
  6. A generalization of the Brauer–Fowler theorem
  7. On groups with large verbal quotients
  8. Conjugacy class numbers and nilpotent subgroups of finite groups
  9. On the common transversal probability
  10. Classifying primitive solvable permutation groups of rank 5 and 6
  11. Tuple regularity and 𝑘-ultrahomogeneity for finite groups
  12. Skew-morphisms of elementary abelian 𝑝-groups
  13. On algebraic normalisers of maximal tori in simple groups of Lie type
https://doi.org/10.1515/jgth-2022-0092

Articles in the same Issue

  1. Frontmatter
  2. Open mappings of locally compact groups
  3. Groups of profinite type and profinite rigidity
  4. On conjugate separability of nilpotent subgroups
  5. Groups whose proper subgroups of infinite rank have a permutability transitive relation
  6. A generalization of the Brauer–Fowler theorem
  7. On groups with large verbal quotients
  8. Conjugacy class numbers and nilpotent subgroups of finite groups
  9. On the common transversal probability
  10. Classifying primitive solvable permutation groups of rank 5 and 6
  11. Tuple regularity and 𝑘-ultrahomogeneity for finite groups
  12. Skew-morphisms of elementary abelian 𝑝-groups
  13. On algebraic normalisers of maximal tori in simple groups of Lie type
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