It is not a surprise for anyone that I'm reviewing a little bit of Abstract Algebra.
Reading Fraleigh's book is proving to be quite a good refresher, especially since my short-term goal is to study Coding Theory and the finer aspects of Coding Theory do involve Algebra (actually, one could say that the finer aspects of Coding Theory, especially Error Correcting Codes, depend quite a lot on Algebra).
Well, while studying Group Theory, one can't avoid to be amazed by some results that are not exactly that intuitive. For instance, I just discovered that the additive group \(\mathbb Z[X]\) of univariate polynomials with integer coefficients is isomorphic to the multiplicative group \(\mathbb Q^+\) of positive rationals.
I thought that this was a relatively easy exercise until I posed the problem to a very competent friend of mine and he had some problems trying to come up with the isomorphism. His failure maybe tells me that this is not as obvious as I once thought and I think that this exercise should be better known, as it can show that very differently looking algebraic structures can be "the same" and the most surprising fact here is that we are not dealing with any pathologically "made up" example!
Another thing that maybe is hard for students to grasp is the concept of a factor group. Knowing that a subgroup is normal iff it is the kernel of some homomorphism of groups is a handy way to memorize and "see" what a normal subgroup is. And students should be shown that there are some quite special ("natural"?) normal subgroups of a group: the center of the group and the commutator subgroup, just as two examples.
The latter one, together with a good explanation of what does it mean for taking factor groups (i.e., taking the elements of the normal subgroup and equating them to the unity), leads to a better understanding of quotient/factor groups. A nice thing to prove (that is short and shows the "mechanics" of factor groups) is the fact that a quotient group is abelian iff the normal group used as the "denominator" contains the commutator subgroup.
Another thing that is worth doing is proving that a normal subgroup \(N\) is maximal in a group \(G\) iff \(G/N\) is simple. The proof of this fact is analogous to the proof of the quotient of a ring being a field iff a ring is factored by a maximal ideal (and that we get an integral domain if the ideal is only prime instead of maximal).
Last, two groups that I feel that should merit way more attention than what I perceive as being given in a first course are the dihedral groups \(D_3\) and \(D_4\). Asking the students to draw the lattice of subgroups of such groups is quite enlightening, as they are "natural" groups that are not abelian (but with all the proper subgroups being abelian). Nice examples to have in mind, without any shadow of doubt.