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Let b(G) denote the largest irreducible character degree of a finite group G. In this paper, we prove that if G is a solvable group which does not involve a section isomorphic to the wreath product of two groups of equal prime order p, and if b(G) < pn, then |G:Op(G)|p < pn.

In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true. MSC:15A15

A necessary and sufficient condition for the existence of the orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with dicyclic group and dihedral group were studied by M. R. Darafsheh and M. R. Ž . Pournaki in press, Linear and Multilinear Algebra and R. R. Holmes and T. Y. Ž . Tam 1992, Linear and Multilinear Algebra 32, 21᎐31 . These authors used a certain permutation structure of these groups to prove the necessary condition. In this article we show that the necessary condition found in these previous works is independent of the permutation structures of these groups. ᮊ

Recommender systems detect users' favorites based on their past behavior and provide them with proper suggestions; however, these systems would encounter problems while dealing with users with low or empty usage data. This issue leads to the most prominent challenge of such systems called cold start. In thispaper, we proposea system based on which a modified discrete imperialist competitive algorithm where tags are clustered using K-medoids algorithm. When a new user logs in and enters his/her tags then the system will suggest just a few sources with the largest weight. Experimental results demonstrate improvement of evaluation criteria for recommender system in comparison with other methods.