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
Consider the real number line \(\mathbb{R}\) and the circle of complex numbers centered at the origin with radius 1 which we denote \(S^1\). The line is a group under \(+\) while the circle is a group under \(\times\). \[\\\]
If you place the line so the origin meets the point \((1,0)\), and wrap it around the circle over and over forever, this describes a homomorphism. Symbolically,
\[f:\mathbb{R}\to S^1, x\mapsto e^{i x}\]
The kernel of \(f\) is the set of all multiples of \(2\pi\). Grab some paper and check that \(S^1\) is a group, that \(f\) is a homomorphism, and the kernel is indeed the multiples of \(2\pi\).
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Related Lesson
Order of an Element in a Group
In Abstract Algebra and group theory, the concept of the order of an element in a group is fundamental. Learn the definition and see some examples.
