Can one prove that two Groups are not homomorphic?

Question

Sorry if I sound too ignorant, I'm barely starting to study abstract algebra.

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While learning about Group theory, I found a video that explains the concept of Group Homomorphisms.

If I understood correctly, it goes something like this: $G$ is a group with the operator $*$ and $H$ is a group with the operator $+$, a homomorphism between these two groups would be a function that maps two elements of $G$ into $H$ in the following way:

$f(x)+f(y)=f(x*y)$

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It seems as a pretty simple definition, in fact, I wonder if one could just construct a function $g$ as a homomorphism of two groups, in other words, defining $g$ as some function such that $g(x)+g(y)=g(x*y)$. However, the lady ends the video saying that "some groups are so different from one another, there are no structural similarities at all", meaning -I think- that there exist pair of groups that are not homomorphic.

Is this the case? Or are all groups homomorphic? Whether they are or not, could someone provide a proof or explanation?

Any thoughts/ideas would be really appreciated!

Answer 1

As others have said, merely having a homomorphism between two groups $G, H$ is not useful. But nonetheless, we can sometimes reveal a lot about a group by looking at what kinds of homomorphisms come in and out of it. Others have already mentioned the matter of embeddings, i.e. when there's an injective homomorphism $\phi: G \to H$. This means that $G$ is in a way "part" of $H$, or more specifically there's a subgroup of $H$ isomorphic to $G$.

Another example of how group homomorphisms tell us things about the structure of a group involves an idea called quotient groups. The principle behind them is that I can take a group $G$ and then identify certain elements of $G$ with each other, and then establish an equivalence relation $\sim$ on $G$ such that $G / \sim$ is a group after inheriting the group operation from $G$. We call these "quotient groups".

It's an elementary result that a group $G$ has a quotient group isomorphic to $H$ if and only if there's a surjective homomorphism $\psi : G \to H$. So in this way, we can use the homomorphisms in and out of $G$ to establish structural properties of $G$. There are many, many, many more examples of this in group theory, but these are some basic ones.

Answer 2

This is one way : If the groups $G,H$ have respective orders $|G|,|H|$ with $gcd(|G|,|H|)=1 $; WOLG$|G|<|H|$, then there cannot be a non-trivial injective homomorphism , so that there is no "embedding" or copy of $G$ in $H$ : if they were, the image $h(G)$ of the supposed homomorphism $h$ would be a subgroup of $H$. But the order of a subgroup must divide the order of the group, which is not possible if $|G|$ does not divide $|H|$.

Answer 3

Mathematicians typically don't use the word "homomorphic" that way, for the reason (as pointed out above) that for any two groups, $G,H$, the $0$-map (or trivial map):

$0:G \to H$ given by $0(g) = e_H$ for all $g \in G$ (map everything in $G$ to the group identity of $H$), always exists.

In any case, the existence, or non-existence, of a (non-trivial) homomorphism between two groups is a poor yardstick of similarity; for distinct primes $p, q$ there is no non-trivial group homomorphism $f: \Bbb Z/p\Bbb Z \to \Bbb Z/q\Bbb Z$, but very few mathmeticans would claim this groups are "dissimilar" (they are both simple cyclic groups).

It is (perhaps) more fruitful to think of a homomorphism $f: G \to f(G) \subseteq H$ as "preserving" partial information about $G$ "inside" of $H$. Some information about $G$ can be lost, as homomorphisms are not always one-to-one (when they are one-to-one, this is a highly desirable situation, as we have a "copy" of $G$ inside $H$, and $H$ may be "easier" to understand than $G$).

Groups can be very different from one another (as opposed to, say, finite-dimensional vector spaces over the reals, which are all rather alike). This is mostly because groups have so few rules, that some wildly disparate sets and operations can still qualify to be groups. Mathematicians have developed classifications, such as order, simplicity, etc. to help distinguish groups by type, and it turns out this is....complicated.