Suraj Singh Shivhare Dr. Ajay Sharma
Abstract:
Symmetry has long served as a fundamental organizing principle in mathematics and the sciences. Group actions provide the formal algebraic language through which symmetry is rigorously expressed, enabling the study of how group elements operate on sets, structures, and spaces. Over the past century, the theory of group actions has expanded from classical geometric origins to a central framework bridging algebra, topology, combinatorics, and modern computational research (Armstrong, 2013; Rotman, 2012). This review synthesizes the foundational elements of group actions including orbits, stabilizers, conjugation, and transitivity and examines their structural implications for group classification, homomorphisms, and normal subgroup analysis (Dummit & Foote, 2004).
