Citation: Alajlan, A.I.; Alghamdi, A.M. Soft Groups and Characteristic Soft Subgroups. Symmetry 2023, 15, 1450. https://doi.org/10.3390/ sym15071450 Academic Editor: Hsien-Chung Wu Received: 21 June 2023 Revised: 13 July 2023 Accepted: 18 July 2023 Published: 20 July 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Soft Groups and Characteristic Soft Subgroups Amlak I. Alajlan 1,2, * and Ahmad M. Alghamdi 1 1 Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al-Qura University, P.O. Box 14035, Makkah 21955, Saudi Arabia; amghamdi@uqu.edu.sa 2 Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia * Correspondence: s44277172@st.uqu.edu.sa or amlak4608@gmail.com Abstract: Group theory is the part of mathematics which addresses the study of symmetry. This paper extends the investigation of the soft group theory which Akta¸ s and Ça˘ gman have defined. We study new concepts in the soft group theory such as the center of the soft group, the kernel of soft homomorphism and soft automorphisms with their basic properties. Furthermore, the concept of soft point groups is introduced and the properties of these soft groups are studied. We state the concept of characteristic soft subgroups of a given soft group. Also, some theorems related to this concept are investigated. We study the characteristic soft subgroups of a given soft point group. The characteristic soft subgroups play a significant part in the study of the soft group theory and are useful for understanding the structure of a soft group and its soft automorphisms. As an application of characteristic soft subgroups, they allow us to identify and study important soft subgroups that are preserved under soft automorphisms. Also, practical applications for our theory can be conducted in future work such as the relation with other disciplines in sciences. Keywords: soft sets; soft groups; soft point groups; soft subgroups; soft homomorphisms; soft isomorphisms; soft automorphisms; characteristic soft subgroups; symmetries 1. Introduction In 1999, Molodtsov [1] proposed the notion of soft set theory to solve complex problems and difficulties which are related to uncertainties in probability theory, fuzzy set theory [2], rough sets [3] and other mathematical tools. In fuzzy sets, every element is assigned a grade of membership function between 0 and 1. However, the selection of an appropriate membership function can be difficult in each particular case, especially when the underlying data are complex or uncertain. The rough sets are based on the idea of approximating a set by two subsets: the lower approximation and the upper approximation. This can be a useful way to handle incomplete or missing information, and it may become very complex when addressing large sets of data or complex relationships between elements and sets. While fuzzy sets and rough sets are beneficial frameworks for handling uncertainty, the concept of soft sets provides a more flexible and powerful approach to handling uncertainty by allowing each element of a set to be associated with a set of parameters that represent a different characterization or attributes. This can provide a more structured and complete representation of uncertainty than fuzzy sets or rough sets. Every fuzzy set and rough set can be represented as a particular case of a soft set, wherein the soft set parameters are defined by the degree of membership or the rough set approximations, respectively. Therefore, the soft sets have rapidly developed into a powerful and versatile framing for addressing uncertainty and have numerous applications in different fields. In 2002, Maji et al. [4] discussed the soft sets in decision-making problems and proposed a method for aggregating soft set parameters to make decisions. In a decision-making problem, a soft set can be used to represent each option or alternative being considered. The soft set parameters can represent the various factors that influence the decision, such as cost, risk or benefit. In 2003, Maji et al. [5] presented several basic notions of the soft set theory. In Symmetry 2023, 15, 1450. https://doi.org/10.3390/sym15071450 https://www.mdpi.com/journal/symmetry