Near-vector spaces determined by finite fields

On the number of near-vector spaces determined by finite fields

“…In 2010, Howell and Mayer classified near-vector spaces over finite fields of p (p is prime ) elements up to isomorphism. They also extended the result to a finite field of p n elements in Theorem 3.9, [3]. This theorem assertes that the number of near-vector spaces V = F ⊕m over a finite field F = GF (p n ) is exactly m + φ(p n −1) n − 2 m − 1 up to the isomorphism in Definition 2.2, where φ is the Euler's totient function .…”

“…(confront Lemma 3.7 in [3]). This motivates us to consider the group G := U (p n − 1)/ p , where the operation is the usual multiplication modulo p n − 1 and p = {1, p, .…”

“…Recall that a (right) near-field is a triple (F, +, ·) that satisfies all the axiom of a skewfield, except perhaps the left distributive law, [3]. By, Theorem 3.5 in [3], any finite dimensional near-vector space (V, A) can be decomposed as…”

“…This theorem assertes that the number of near-vector spaces V = F ⊕m over a finite field F = GF (p n ) is exactly m + φ(p n −1) n − 2 m − 1 up to the isomorphism in Definition 2.2, where φ is the Euler's totient function . This number is calculated based on the number of distinct suitable sequences, (Definition 2.3 in [3]). Namely, if A * 1 and A * 2 are determined by suitable sequences (S 1 ) and (S 2 ), respectively, then (F ⊕m , A * 1 ) ∼ = (F ⊕m , A * 2 ) if and only if (S 1 ) = (S 2 ).…”

“…This contradicts to the main results of the paper. A slip can be found in the proof of Theorem 3.9 in [3] (line 17th in the proof) in which there is using the isomorphism η to be η(s α ) = t α . In fact, this should be η(s α ) = t α q , for some 1 ≤ q ≤ p n − 1 and gcd(q, p n − 1) = 1.…”