Some clarifications about the ideas of representation of a group

I started to study group theory but I have many doubts about the topic, so I'd like to share my current understanding together with some questions, my aim is to understand the general ideas and concepts about the topic more than the specific calculations, so answers with little math will be appreciated.

I've seen that a representation of a group is a homomorphism, that means: let $$G$$ and $$H$$ be two groups, and $$f$$ a map from $$G$$ to $$H$$ such that $$\forall g\in G \Rightarrow f(g)\in H$$. Then $$f$$ is a homomorphism if $$\forall g_1,g_2\in G \Rightarrow f(g_1g_2)=f(g_1)f(g_2)$$. Now what I understood about this is that a representation conserves the information of the operation inside the group, so $$G$$ and $$H$$ may be different but the operations of the two groups follow the same rules. Does it mean that if I know just the operation rules of $$G$$ and $$H$$ I can't distinguish them? If yes, can you give me an example of what these abstract rules are?

Then an isomorphism is an homomorphism that is also bijective, which means that if two groups have an isomorphism, one is equivalent to the other and the only way to distinguish them is to look at their elements. So, if a representation would be an isomorphism, I would understand its meaning as another way to express the same group. But, since it's just an homomorphism, I'm struggling in understanding why it's useful to know the representations of a group.

• Would Mathematics be a better home for this post? Also please only ask 1 question per post. – Qmechanic Apr 27 '20 at 10:07
• I was about to write that. Users will feel a bit discouraged to answer if they have an answer for just a subset of the questions. Hence the website limits one question per post. – MannyC Apr 27 '20 at 10:09
• Hi Simo Bartz, I removed the second question. – Qmechanic Apr 27 '20 at 16:11

Groups can be anything from abstract spaces to for example the vector spaces we all know such as $$\mathbb{R}^n$$. A homomorphism allows us to compare groups and map certain properties from one to the other. For example, consider the group $$\mathbb{Z}$$,+ (the group of integers equipped with the standard sum). We can define a function $$\phi: \mathbb{Z},+ \to \mathbb{Z}_5,+: a \mapsto a \mod{5}$$ for example, with $$\mathbb{Z}_5$$ being the multiplicative group modulo 5 and $$\phi$$ in this case the canonical map. Modulo groups are very prominent in group theory so you should get to know them! In this case, the operator + is the same for both groups but they are indeed very different and are distinguishable (for example $$\mathbb{Z}$$ has infinite elements while $$\mathbb{Z}_5$$ only has 5). You can check with the definition you included that this is a homomorphism. For example: $$\phi(2+6) = \phi(8) = 3$$ and also $$\phi(2)+\phi(6) = 2 + 1=3$$.
From this arrives the question: what about groups that are more similar to eachother? For this we use the isomorphism. For the homomorfism to be bijective, this would also include for example that the amount of elements must be equal. When a new group is encountered, existing properties of a different group can be instantly applied if it is known that an isomorphism exists. An isomorphism guarantees that the groups "behave" in the same way you could say. In our previous example, you could find all information about $$\mathbb{Z}_5$$ coming from $$\mathbb{Z}$$ but not the other way around. The isomorphism gives a path in both directions. An example of an isomorphism would be $$\psi: \mathbb{Z}_3 \times\mathbb{Z}_2 \to \mathbb{Z}_6:(a,b) \mapsto (3a + 4b) \mod{6}$$, you can check this out in the wikipedia I linked above. We can in fact get all information of the multiplicative group of any number by "only" using information out of the multiplicative groups of its factors in the prime factorisation. It is a one-to-one map. The inverse can be found using the opposite direction of the maps in that wiki-page (which might not be a "normal" function but just a map from point to point).