9: The Isomorphism Theorem
Recall that our goal here is to use a subgroup of a group \(G\) to study not just the structure of the subgroup, but the structure of \(G\) outside of that subgroup (the ultimate goal being to get a feeling for the structure of \(G\) as a whole). We've further seen that if we choose \(N\) to be a normal subgroup of \(G\), we can do this by studying both \(N\) and the factor group \(G/N\). Now, we've noticed that in some cases—in particular, when \(G\) is cyclic–it is not too hard to identify the structure of a factor group of \(G\). But what about when \(G\) and \(N\) are more complicated? For instance, we have seen that \(SL(5,\mathbb{R})\) is a normal subgroup of \(GL(5,\mathbb{R})\). What is the structure of \(GL(5,\mathbb{R}) / SL(5,\mathbb{R})\)? That is not so easy to figure out by looking directly at left coset multiplication in the factor group.
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- 9.1: The First Isomorphism Theorem
- A very powerful theorem, called the First Isomorphism Theorem, lets us in many cases identify factor groups (up to isomorphism) in a very slick way. Kernels will play an extremely important role in this. We therefore first provide some theorems relating to kernels.