On fundamental isomorphism theorems in soft subgroups R. Barzegar a∗ ; S. B. Hosseini b † a,b Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran. Abstract In this paper, by the use of soft sets and also their duality, we introduce the new conceps on soft sets , which are called soft quotient subgroup and quotient dual soft subgroup. Then we derive algebraic proprties about them and in sequel, investigate the fundemental isomorphism theorems in soft subgroups analogous to group theory. Keywords: Soft subgroup; Normal soft subgroup; Homomorphism of soft subgroups; Quotient–terms. 1 Introduction Most of our traditional tools for formal modeling, reasoning, and computing are crisp, deter- ministic, and precise in character. But many complicated problems in economics, engineering, environment, social science, medical science, etc., involve data which are not always all crisp. We cannot always use the classical methods because of various types of uncertainties present in these problems. As a new mathematical tool for dealing with uncertainties , incomplete information and the study of intelligent systems characterized by insufficient control systems, for the reason, in 1965 the notion of fuzzy sets was introduced by L. Zadeh [21] and in 1982 the concept of rough set theory which proposed by Z. Pawlak [16, 17] and Z. Pawlak and A. Skowron [18]. But each of these theories has its inherent difficulties as pointed out by D. Molodtsov [15] in 1999. D. Molodsov [15] initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. Soft set theory has potential applications in many different fields some of them are the algebraic structure of soft sets [1, 2, 3, 6, 7, 8, 9, 10, 19, 20] and decision making [5, 14]. Application of soft sets to groups was introduced by H. Aktas and N. Cagman [2]. To extend the soft set in group theory, N. Cagman et al. [4, 5] introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In the paper, we use and review the theory and present about soft qoutient subgroup and dual of soft subgroup and give some results and review the fundemental homomorphism theorems in soft subgroups. The rest of the paper is organized as following: In section 2, we review some basic concepts and properties of soft sets and soft subgroup. In section 3, we study on soft quotient subgroup and quotient soft subgroup. In section 4, we investigate to quotient dual soft subgroups and fundamental theorems. Conclusions are given in Section 5. ∗ Corresponding author; Email address: ro.gbps@gmail.com † Email address: sbhosseini@iausari.ac.ir 1