Simpler description of isomorphic groups property

Question

In order to show that two groups are isomorphic, according to my book, we need to find a mapping function that is one-to-one and also onto, a bijective function called $\phi$ that maps elements from one group to another.

However, that is not enough to make sure the resulting Cayley table for the second group is actually valid. We also need to make sure that for two groups $(G, \circ)$ and $(H, *)$ the following property holds:

$$\phi(x \circ y) = \phi(x) * \phi(y)$$

For all $x, y \in G$.

I understand what this means using symbols.

But what I'm looking for is some description of this property in plain English (does this property have a name?), so to make it more easier for me to remember and digest.

My best effort would be something like: all elements $x, y$ (and also $x \circ y$) of $G$ must have a valid representation in $H$.

But that's a little too vague to be useful. Can someone expand to describe this property so that I could explain it to someone else using my own words? Otherwise I feel I'm struggling to really get this into my head.

Answer 1

Such a map is called a groups homomorphism, so the property you're talking about is the homomorphism property. A homomorphism between two sets with a given structure is a map which preserves the structure, in this case, group structure, i.e. the image of a product is the product of the images: it preserves the only operation of groups, multiplication.

Answer 2

You need a bijection to identify the elements (this makes sure that the elements of one of the groups are only a renaming of the elements of the other group. But this does obviously not tell you anything about the group structure. To get something that behaves like the same group, you need some extra conditions which also tells you that the elements also multiply in the same way after renaming them. That is exactly the group homomorphism condition (at least for bijective maps, otherwise there might be things in the kernel for example or there are things that did not come from renaming). So you could say that a group isomorphism is a renaming where you keep/transfer the multiplication table according to the bijection.

Maybe take two groups of order $2$ (for example $\lbrace 1,-1 \rbrace$ with the standard multiplication of numbers and $\mathbb{Z}/2\mathbb{Z}$) and write down the multiplication tables to see that they look the exact same. This will show you how to define a group isomorphism then by sending the elements to the element that behaves like it in the other group. Since you cannot write down multiplication tables all the time (its messy and would be quite difficult for infinite groups in general), we need this other way of identifying when two groups are the same up to renaming. That is the existence of a group isomorphism. I like to say that two isomorphic groups are basically a person in his regular clothes and the same person in a full-body costume on halloween. You might not recognize him at first, but he is the same person even when he looks differently.