In this final section of Chapter 9 Hungerford discusses the Structure of Finite Groups. In the previous sections it has been mentioned that groups of order four are isomorphic to or and in this section Hungerford connects this idea to larger groups of prime order or order where p is prime.
Conjugacy used here in Section 9.4 seems to be at least partially related to normal groups. For normal groups and does not necessarily equal . Here a is said to be conjugate to b if there exists an such that . There’s a distinct similarity that may have some significance, or we may be able to reword […]
This section is mostly about the Sylow Theorems. While these theorems assist in analyzing nonabelian finite groups Hungerford lacks examples in explaining these theorems. The biggest question then is how do the Sylow Theorems help understand nonabelian finite groups?
In this section Hungerford classifies all Finite Abelian Groups. The first section that may cause some confusion is in Theorem 7.9 where he suddenly calls upon k without previously defining it. It seems that k may be the the number such that (in the multiplicative notation) or (in the additive notation). The Fundamental Theorem of Finite Abelian […]
Thus far Direct Products have been mentioned only as a way to use various groups or rings. In this section Hungerford says that the Direct Product of a group (or groups) can tell us much about the group itself. How is this possible? The Direct Product of any number of groups operate only on the […]
This section on the Simplicity of doesn’t seem to say a lot unless fully understood. The part easiest to understand is that only the group is not simple. To understand we must understand what they are.
Once again the parallels between groups and rings is shown here. Possibly one of the most interesting parts of this section is the First Isomorphism Theorem. This is an idea that has appeared a few times (as rings and every now and then hints of this have appeared in this section on groups).
Once again the parallels to rings return in this study of groups. Here Hungerford explains Quotient Groups. They have properties that one might expect such a group to have. The coset notation is probably the most difficult part of this section. Properties of groups have thus far been seen to carry into subgroups and so […]
While this comment could have been absorbed into the previous comment on this section, it is important to note that Hungerford specifies that does not imply that for every when dealing with groups and normal groups.
This section is rather interesting. There exist normal subgroups which have cosets which resemble abelian subgroups.