Section 16.1 Pre-read
Recall that an isomorphism between groups \(G_1,G_2\) is an operation-preserving bijective correspondence between the two groups.
What if we relax the bijective correspondence bit?
Definition 16.1.1. Homomorphism.
An operation-preserving mapping \(\funcdef{\varphi}{G_1}{G_2}\) from one group to another.
You will be asked to look at a number of examples of homomorphisms in the Discovery guide following this Pre-read section.
Warning 16.1.2.
Since a homomorphism is not necessarily a bijective correspondence, we must now keep the following in mind.
It may be possible for two (or more!) elements in \(G_1\) to correspond to the same element in \(G_2\text{.}\) That is, we might find \(\varphi(x) = \varphi(y)\) to be true for some \(x \neq y\) in \(G_1\text{.}\)
There may exist elements in \(G_2\) that have no corresponding element in \(G_1\text{.}\) That is, we may find examples of an element \(z\) in \(G_2\) for which \(\varphi(x) = z\) never occurs.
Remark 16.1.3.
With this definition of homomorphism now in hand, an isomorphism becomes a special type of homomorphism. In particular, an isomorphism is a bijective homomorphism.
Definition 16.1.4. Image of a homomorphism.
The collection of all possible outputs of \(\varphi\) in \(G_2\text{.}\) That is, the collection of all elements \(z\) in \(G_2\) for which it is possible to find at least one example of a corresponding element \(x\) in \(G_1\) with \(\varphi(x) = z\text{.}\)
We write \(\im \varphi\) or \(\varphi(G)\) to denote the image of \(\varphi\text{.}\)
Definition 16.1.5. Kernel of a homomorphism.
The collection of all inputs in \(G_1\) that are mapped to the identity element in \(G_2\text{.}\) That is, the collection of all elements \(x\) in \(G_1\) so that \(\varphi(x) = e_2\text{,}\) where \(e_2\) represents the identity element in \(G_1\text{.}\)
We write \(\ker \varphi\) to denote the kernel of \(\varphi\text{.}\)
Warning 16.1.6. Keep it straight!
The image \(\im \varphi\) is a subcollection of \(G_2\text{,}\) while the kernel \(\ker \varphi\) is a subcollection of \(G_1\text{.}\)
Remark 16.1.7.
There was no need to define either concept of image or kernel when we studied isomorphisms, because, as an isomorphism is always a bijective correspondence, the image of an isomorphism \(G_1 \to G_2\) is always the full group \(G_2\text{,}\) and the kernel always consists of just \(e_1\text{,}\) the identity element in \(G_1\text{.}\)