This final section of 8.1 deals with finite groups and their structures. This section is about finding isomorphisms between finite groups.
Section 8.1 has many parallels to the material previously seen in Hungerford’s book. Union of cosets (Theorem 8.4) and the notation used in Lagrange’s Theorem (|G|=|K|[G:K]) are new. These two are also some of the most difficult concepts. What does this new notation mean?
In this section Hungerford introduces cycle notation. Though it is an simple way to write groups compared to the recently introduced notation, it requires practice to assimilate. Many times Hungerford mentions permutations, but what does it mean for to be both even and odd?
This section contains much of the same principles of isomorphism as rings. There are a few ideas that are new for example: If G is abelian and H is nonabelian, then G and H are not isomorphic.
The idea that a subgroup can generate the original group is an interesting idea. The definition of a subgroup doesn’t mention anything about infinite subgroups, so they should be a possibility. (Later they are explicitly mentioned but only with cyclic subgroups.)
This section is fascinating. It establishes some of the basic properties of groups, but the one that I find particularly interesting is in Theorem 7.8 on the order of elements. Each element may be the identity element if acted upon by itself enough times
In the first part of Section 7.1 there was an emphasis that Groups only have one operation, addition. Yet here we see that there is another operation, multiplication. So why does Hungerford go through all the trouble of defining addition as the only operation in a group when a page later this changes?
Groups are fascinating systems. Unlike rings they only have one operation. Their notation, however, seems to be difficult to write in the form of functions – at least up to the end of the first example it seems like this. After some brief reading online it seems that this has a lot of application, much …
These types of functions are vey important to understand. They describe how the function preserves distinctness, and how the functions interact with both the codomain and the domain. Injective
This section on “The Structure of R/I when I is Prime or Maximal” provided a lot of information in very small number of pages. It defined what it means for a commutative ring to be prime and what a maximal is. In defining the former Hungerford explains “[q]uotient rings were developed as a natural generalization of the …