The theory of group representations deals with the classification of homomorphisms of the abstract group G into linear transformation groups or matrix groups. This paper develops the theory of representations of finite groups in finite-dimensional vector spaces. The field K over which vector spaces are defined is often (not always) taken as complex numbers. Mathematical interpretation is at a level between representation theory for physicists and representation theory for mathematicians. In the first section of the first chapter, we introduce the basic definitions of group representation theory and provide examples to illustrate them. The theory of representations of finite Abelian groups is presented in full, and we are also talking about the characters of representations, Schur’s lemma, etc. In the second chapter, we study irreducible representations of the symmetric group Sn based on the Young symmetrizer technique. The third chapter discusses a new look at obtaining a Young symmetrizer for a symmetric group.