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Journal of Discrete Mathematical Sciences and Cryptography

(Since 1998)
Hybrid ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

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Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

Semilinear groups contained in alternating group on a finite-dimensional linear space

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pp. 1051–1062Vol. 26Issue 4June 2023DOI: 10.1080/09720529.2021.1974649 Crossmark XML
Received:
01 May 2021
Published Online:
08 Mar 2022
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-1450
Pages:
1051–1062

Abstract

Let  be a finite field with q elements, and n a positive integer with n ≥ 2, where q = pm (p : a prime, m : a positive integer). It is natural to ask under which conditions a semilinear group over  is a proper subgroup of the alternating group . In this paper, we provide necessary conditions for semilinear groups over  to be subgroups of the alternating group  on the n-dimensional linear space . We shall show that if q =2 and n ≥ 3, or p =2 and n ≥ 2, then the affine semilinear group  (resp. the general semilinear group ) is a proper subgroup of the alternating group , respectively. In addition, we shall also prove that if one of the following five conditions holds: (i) q =2 and n ≥ 3, (ii) p =2 and n ≥ 2, (iii) p ≡ 1 mod 4 and n ≥ 3, (iv) p ≡ 3 mod 4 and m : odd, (v) p ≡ 3 mod 4, m, n : even, then the affine special semilinear group  (resp. the special semilinear group ) is a proper subgroup of the alternating group Alt , respectively. Our results are generalizations and improvements of several known results. These results can be derived from the formula of the sign of the permutation induced by the pk -th Frobenius maps on  (1 ≤ k ≤ m).

Keywords

Alternating groupFrobenius automorphismSemilinear groupSemilinear transformation

Subject Classifications

(2010) Primary 15A04Secondary 12E2020B2520D06

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