Building intuition in group theory

Question

I'm finding it hard to translate abstract results of group theory into something that intuitively makes sense.

Putting this into a concrete example: if $f:G\to H$, $Im(G)$, is a subgroup of $H$? Is there an intuitive reason why this should be so?

More generally, does anyone have any tips for helping to 'understand' a result in a deeper sense than simply knowing that it's true?

Thanks!

Answer 1

When we come across any kind of mathematical/algebraic object, one of things we should define is the notion structure preserving map between two such objects. Such a map should preserve the key properties of the structure.

A homomorphism of groups is a function that preserves the key structure of the group. Think about the conditions for $\theta:G\to H$ to be a homomorphism: we have

1. $\theta(e_G) = e_H$ - i.e. $\theta$ takes the identity in $G$ to the identity in $H$
2. $\theta(g*g')=\theta(g)*\theta(g')$ - i.e. $\theta$ preseves multiplicaton

As a corollary of (1) and (2) we can also show that

3. $\theta(g^{-1}) = \theta(g)^{-1}$ - i.e $\theta$ preserves inverses

In other words, the map $\theta$ preserves the defining structure of our group. As such, it makes sense that $\operatorname{Im}(\theta)$ should also be a group, since it inherits its group structure from $G$.