We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with Semi-Dihedral groups SD8n. The dimensions of these symmetry classes of tensors are also computed. MSC(2000):Primary 20C30; Secondary 15A69

Abstract. A slip on a paper concerning near-vector spaces is fixed. New characterization of near-vector spaces determined by finite fields is provided and the number (up to the isomorphism) of these spaces is exhibited.

A generalized matrix function d G χ : M n (C) → C is a function constructed by a subgroup G of S n and a complex valued function χ of G. The main purpose of this paper is to find a necessary and sufficient condition for the equality of two generalized matrix functions on the set of all symmetric matrices, S n (C). In order to fulfill the purpose, a symmetric matrix S σ is constructed and d G χ (S σ ) is evaluated for each σ ∈ S n . By applying the value of d G χ (S σ ), it is shown that d G χ (AB) = d G χ (A)d G χ (B) for each A, B ∈ S n (C) if and only if d G χ = det. Furthermore, a criterion when d G χ (AB) = d G χ (BA) for every A, B ∈ S n (C), is established.

Abstract. This paper provides some properties of Brauer symmetry classes of tensors. A dimension formula is derived for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups whose non-linear Brauer characters have support being a cyclic group. Using the derived formula, necessary and sufficient condition are investigated for the existence of an o-basis of dicyclic groups, semi-dihedral groups, and also those things are reinvestigated on dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included.