I have been working on the problems on homomorphism, the problem encountered is
From the additive group $\mathbb{Q}$ to which one of the following groups does there exist a
non-trivial group homomorphism?
(A) $(\mathbb{R},×)$, the multiplicative group of non-zero real numbers.
(B) $\mathbb{Z}$ , the additive group of integers
(C) $\mathbb{Z_2}$, the additive group of integers modulo 2
(D) $(\mathbb{Q},×)$, the multiplicative group of non-zero rational numbers.
For option A ,I could directly think of $$ f: (\mathbb{Q},+) \rightarrow (\mathbb{R},×) $$ as $$f(x)=e^x ; x \in \mathbb{Q} $$
Which is a homomorphism and being the question is single correct,I was done. But I am unable to claim that other options are incorrect, and I thought that if I am unable to find a function, that doesn't mean that it "does not exist".
There are several ways to prove the given function is a homomorphism but I want to find the ways to claim that the given groups are not homomorphic. I can think of the following;
1 Homomorphic image of a cyclic group is cyclic.(it's counter statement, that if co-domain group is non cyclic and domain is cyclic then no "non-trivial homomorphism".
2 Under some homomorphism from a group G to G', the normal subgroup H of G, can be made kernel. ( From this it can concluded that if I map a normal subgroup H to Identity element of G' then that that mapping should be a homomorphism(having doubt in this statement)).
And this would be very long process and it's based on just hit and trials.
Question
Is there any systematic way to claim that there does not exist any non trivial homomorphism between the given two groups?
Best Answer
I guess the "systematic way" that you are looking for to show that there is no nontrivial homomorphism $G \to H$ is to identify a group-theoretical property $\mathcal{P}$ that is preserved under group homomorphisms, which $G$ has but which no nontrivial subgroup of $H$ has.
As kabenyuk pointed out in comments, $G$ has the property (called divisibility) that, for every $g \in G$ and every integer $n>0$, there exists $h$ in $G$ with $nh=g$.
No nontrivial subgroup of any of the groups in (B), (C) and (D) has this property. This is clear for (B) and (C), and any nontrivial subgroup of $({\mathbb Q}\setminus \{0\},\times)$ has an element with no $n$-th root fpr some $n$.