Preserving an Algebraic Structure

Question

What does it mean that a homomorphism preserves algebraic structure of a group ? What does it mean algebraic structure? Can someone please explain in simpler and detailed terms.

Answer 1

The definition of a homomorphism $\varphi:G \to H$ for groups $G, H$ is given by the following axiom:

$\varphi(gg') = \varphi(g)\varphi(g')$ for $g,g' \in G$,

It can be shown that $\varphi(1_G) = 1_H$, the units of $G$ and $H$ respectively and $\varphi$ maps inverses to inverses by $\varphi(g^{-1}) = \varphi(g)^{-1}$

What do all of these mean? In the first axiom, our homomorphism takes the composition of any two elements in $G$ and sends it to the composition of two elements in $H$, $\varphi(g)$ and $\varphi(g')$. Furthermore, it preserves identity: $1_G$ is mapped to $1_H$ and it preserves inverses.

Hence, we can see that the homomorphism maintains the "group structure" by, in a sense, preserving the law of composition, identities, and inverses; the homomorphism is "translating" the structure of a group G to the structure of another group H.

When there exists a bijective homomorphism, an isomorphism, between two groups then we can consider the two groups nearly the same as they have the same properties; the homomorphism translating the properties of the first group to the ones of the second can be inverted, so there is little distinction between the two groups.

Disclaimer: I don't claim to be an expert, just throwing in my own insights