How to determine non trivial homomorphisms

Question

I am trying to understand and it doesn't make any sense to me:

How can I determine if there are any non trivial homomorphisms between groups or rings? How do I find them? and once I found them, how can I know that there aren't any more of them?

Answer 1

Of course you must have some insider information to construct homomorphisms. One case is: one of the groups is a direct product of many groups with one factor a cyclic group of order $n$, and the other group has an element of order a divisor of $n$.

That is $G_1=C_n\times H$ and $G_2$ has a cyclic subgroup of order $m$, with $m|n$.

In this situation we have a homomorphism from $G_1$ to $G_2$ that sends the cyclic group $C_n$ of order $n$ to the aforementioned subgroup of $G_2$ order $m$. (And all elements of $H$ will be in the kernel,and some more possibly in $C_n$ too).

Answer 2

Your question is broad, and I am going to reply with an equally broad answer.

Understanding the homomorphic images of a group is the same as understanding its quotients. That is, a group homomorphism $\varphi:G\to H$ is the same as the projection $G\to G/\ker\varphi$ followed by an inclusion $G/\ker\varphi\hookrightarrow H$. This is a consequence of the first isomorphism theorem. Therefore, asking whether there is a homomorphism from $G\to H$ is the same as asking whether there is a quotient $G/N$ (and a normal subgroup $N$) of $G$ which is isomorphic to a subgroup of $H$.

So to understand the homomorphisms from one group to another, you have to understand the quotients of the first group and the subgroups of the second.

(Everything I just said is true with "ring" replacing "group" and "ideal" replacing "normal subgroup".)

Answer 3

I will try to answer this as well as I can from the perspective of group homomorphisms (I don't really know anything about rings, but I imagine much of this translates over). First let's look at the definition:

Let $G,H$ be groups, and let $\phi:G \to H$ be a function, such that for all $x,y \in G$, $$\phi(xy)=\phi(x)\phi(y)$$ where we use the group operations of $G$ and $H$ in the obvious places.

What exactly does this mean? What we're saying is doing $\phi$ to elements of $G$ keeps some of the structure of $H$. That is, in some sense, elements of $G$ behave in a similar way to elements of $H$. For example, let $G$ be $\mathbb{R}$ under addition, and let $H$ be the positive reals under multiplication. Then the homorphism $\phi(x)=e^{x}$ is a homomorphism from $G$ to $H$ because $$\phi(x+y)=e^{x+y}=e^{x}e^{y}=\phi(x)\cdot\phi(y)$$

So, essentially, looking for homomorphisms boils down to trying to identify how two groups behave in similar ways (if they behave in exactly the same way, then your homomorphism will turn out to be a bijection, in which case we call it an isomorphism).

EDIT: In the comments you ask how many homomorphisms there are from $\mathbb{Z}$ to itself. Continuing the theme above, let's try to capture the structure of $\mathbb{Z}$. It is the infinite cyclic group with one generator: we take our single generator, $1$, and take all its copies and its inverse and its copies. Well, all we need to do is pick a generator: any integer will do, and no two integers will generate the same cyclic group (try to prove this).