Given a finite group G of cardinality N, the group determinant [theta]G associated to G is a homogeneous polynomial in N variables of degree N. We study two properties of [theta]G. First we determine the stabilizer of [theta](G) under the action of permuting its variables. Then we also prove that the Lehmer's constant for any finite abelian group must satisfy a system of congruence equations. In particular when G is a p-group, we can strengthen the result to establish upper and lower bounds for the Lehmer's constant. / text

Linear characters of finite groups can be extended to take k operands. The basics of such a k-fold extension are detailed. We then examine a proposition by Johnson and Sehgal pertaining to these k-characters and disprove its converse. Probabilistic models can be applied to random walks on the Cayley groups of finite order. We examine random walks on dihedral groups which converge after a finite number of steps to the random walk induced by the uniform distribution. We present both sufficient and necessary conditions for such convergence and analyze aspects of algebraic geometry related to this subject.

The aim of this work is to analyse functions of French articles as a part of the text in comparison with the means of Czech language. The theoretical part defines determinants as a word class in French language as well as its function in conjunction with nominal group, ways of linking in the text and structure of the text. All of this is based on the works of French and Czech authors concerned in the question of articles. The comparative part was done on the text accessible in both languages. The texts were analysed in the same way and then they studied all the questions from the theoretical part i.e. functions of the article in the text and its language means in Czech. The results were put into two tables. First table presents Czech equivalents of nominal group. Second table presents Czech equivalents of the way of linking in the text.