Let R>0 be the group of positive real numbers under multiplication. Let CX be the group of nonzero complex numbers under mu!tiplication. Let S1 = {a + bi such that a^2 + b^2 = 1) be the subgroup of C consisting of all complex numbers of absolute value 1. Note that is normal in Cx since Cx is abelian. Prove that CX/S1 is isomorphic to R>0. [Hint: Find a homomorphism from Cx to R>0 and use the First Isomorphism Theorem.]

Homomorphism and First Isomorphism Theorem are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

... Let ker(phi) denote the kernel of phi. [The first isomorphism theorem states that if phi is a homomorphism from a group G onto H, (note that it must be onto ...

... way I usually approach these problems is to try and use the isomorphism theorem, where is a homomorphism. The most "obvious" homomorphism is is First we must ...

... to G. If G has elements x and y , each of order 4, with x2 = y 2 , can you determine G (up to isomorphism)? ...First, I show that f is a homomorphism. ...

... where N is the corresponding kernel, are isomorphic. ...Homomorphisms, Kernels, Isomorphisms and Fields are investigated ... use definitions from "A First Course in ...

... det X = 1} = SLn(k) As the kernel of a group homomorphism, SLn(k ... of GLn(k)). Hence, det is surjective, and according to the first isomorphism theorem for groups ...

... (c) The nonzero quaternions H^x are isomorphic to a subgroup of GL2(C) via the map. ... is a ring monomorphism (1-1 homomorphism). To see this, we first note that. ...

... as a set of all permutations on the first n integers ... All cyclic groups of an order n are isomorphic to the ... mapping f:SG', there exists a unique homomorphism g:GG ...

... Since phi_i is surjective, according to the first group homomorphism theorem, P/ker(phi_i) is isomorphic to A_i and thus A_i is a quotient group of P. (c) I ...

... This shows that is a homomorphism. ... is a multiple of n. This proves that is an isomorphism. ... The first characterizes the group of integers modulo n in ...