One can also note that Abelian groups G have the property that GG,which non-Abelian groups do not enjoy. Note A good(and concise)reference for group representation is [Ser77].[CvD10]also has an appendix for some basic knowledge about group representation,though it takes a matrix (instead of operator)view.For general introduction of algebra,see [Art10].For more extensive introduction of abstract algebra,a standard textbook is [DF03]. Reference [Art10]Michael Artin,Algebra,2 edition,Pearson,2010. [CvD10]Andrew M.Childs and Wim van Dam,Quantum algorithms for algebraic problems,Reviews of Modern Physics,Volume 82,January-March 2010 [DF03]David S.Dummit and Richard M.Foote,Abstract Algebra,Wiley,2003. [Ser77]Jean-Pierre Serre,Linear representations of finite groups,Springer-Verlag,1977. Exercise 1.Prove that characters are class functions,namely x(xyx-1)=x(y).Also prove another property: x(x-1)=X(x)* 2.Check that the Fourier transform defined is unitary.One can also note that Abelian groups 𝐺 have the property that 𝐺 ≅ 𝐺̂, which non-Abelian groups do not enjoy. Note A good (and concise) reference for group representation is [Ser77]. [CvD10] also has an appendix for some basic knowledge about group representation, though it takes a matrix (instead of operator) view. For general introduction of algebra, see [Art10]. For more extensive introduction of abstract algebra, a standard textbook is [DF03]. Reference [Art10] Michael Artin, Algebra, 2 nd edition, Pearson, 2010. [CvD10] Andrew M. Childs and Wim van Dam, Quantum algorithms for algebraic problems, Reviews of Modern Physics, Volume 82, January–March 2010. [DF03] David S. Dummit and Richard M. Foote, Abstract Algebra, Wiley, 2003. [Ser77] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, 1977. Exercise 1. Prove that characters are class functions, namely 𝜒(𝑥𝑦𝑥 −1) = 𝜒(𝑦). Also prove another property: 𝜒(𝑥 −1) = 𝜒(𝑥) ∗ . 2. Check that the Fourier transform defined is unitary