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

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).