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 ...

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

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... (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. ...