Section 8.1 Part 1: Analysis

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?

Section 7.5: Analysis

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?

Section 7.3: Analysis

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.)

Section 7.2: Analysis

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

Section 7.1: Part II Analysis

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?