Complete purely algebraic proof of the homomorphism between SU(2) and SO(3) without concerning their topological properties

The aim of this paper is to provide a complete purely algebraic proof of homo-morphism between SU (2) and SO(3) without concerning the topology of bothgroups. The proof is started by introducing a map ϕ : SU (2) → M L(3, C) de-fined as [ϕ(U )] i j ≡ 12 tr(σ i U σ j U † ). Firstly we proof that the map ϕ satisfies[ϕ(U 1 U 2 )] i j = [ϕ(U 1 )] i k [ϕ(U 2 )] k j , for every U 1 , U 2 ∈ SU (2). The next step is toshow that the collection of ϕ(U ) is having orthogonal property and every ϕ(U ) hasdeterminant of 1. After that, we proof that ϕ(I 2 ) = I 3 . Finally, to make sure thatϕ is indeed a homomorphism, not an isomorphism, we proof that ϕ(−U ) = ϕ(U ),∀U ∈ SU (2).