In case when a particular mapping $\phi$ is not given.
I just want to know what are the steps to be employed while proving homomorphism.
If $\phi$ is given, then one can prove the homomorphism as per the rules of definition. But my problem arises when no such mapping is given in the question.
Thanks in advance.
EDIT: To prove there exists a group homomorphsim from $\mathbb{C}$ to $\mathbb{R}$.
On
Finding a group isomorphism between $\mathbb{C}$ and $\mathbb{R}$ is tricky and possibly not what you should be thinking about in the begginings of an introductory class to abstract algebra.
Anyway, consider them as vector spaces over $\mathbb{Q}$. A basis (which exists by axiom of choice) for either of these spaces necessarily has cardinality $2^{\aleph_0}$. It can't be bigger: this is the cardinality of $\mathbb{C}$ and $\mathbb{R}$. And it can't be smaller: with less than $2^{\aleph_0}$ elements and the $\aleph_0$ elements of $\mathbb{Q}$, you can't produce, using only finite linear combinations, the $2^{\aleph_0}$ elements of $\mathbb{R}$ or $\mathbb{C}$.
Take a bijection between these bases and this gives an isomorphism as vector spaces, which is particularly an isomorphism as groups.
I just noticed you only want a homomorphism, which can be found by simpler means. But maybe this answer is worth posting anyway.
You must construct one! Although you may look at different properties isomorphic groups have. Like Order of the group and orders of elements . Subgroups have the same properties too. If its a cyclic -abelian or not. And otheproperties. The definition says if there exists a map that.... .So you have to find a map. Ive never met a proof where it does prove its existance of this map .Whitoutfinding it. Now there are other questions like. THere CANT be homomorphisms doing "that" or some other thing. Those questions are answered by using known facts and properties of the Homomorphsim. Combined with the structure of the given groups you get restrictions on the possible homomorphisms. EDIT : This was an answer before the Question became finding a specific homomorphism.