## Discussion

The ** kernel** of a group homomorphism measures how far off it is from being one-to-one (an injection). Suppose you have a group homomorphism: \[f:G\to H\]

The kernel is the set of all elements in \(G\) which map to the identity element in \(H\). You can write this using mathematical notation as:

\[ \text{ker}(f) = \{x\in G | f(x)=e\} \]

It is a subgroup in \(G\), not \(H\), and it depends on \(f\). Different homomorphisms between \(G\) and \(H\) can give different kernels.

If \(f\) is an *isomorphism*, then the kernel will simply be the identity element. You can also define a kernel for a homomorphism between other objects in abstract algebra: rings, fields, vector spaces, modules. We will cover these in separate separately.

## Example / Exercise

Real Number Line & the Circle

## Related Lesson

Order of an Element in a Group