Isomorphisms and Cosets

Just an hour ago, I had my most recent meeting with Dr. Prudhom, in which we discussed several exercises from the chapter on isomorphisms and several more from the chapter on cosets and Lagrange’s theorem. Most of them were fairly straightforward, except for several that were a bit trickier—there was even one that we still haven’t figured out.

I’ll begin with the ones from the isomorphism chapter, which were overall easier than the coset exercises. The first group of exercises that I tackled formed a little sequence, where each one built on the previous ones:

58 was not particularly difficult. A solution very similar to the one that I found can be found in the answer to this Math Stack Exchange question. 59 was tricker since it required that I find a specific example of a proper subgroup of the rationals that is isomorphic to the positive rationals under multiplication. I ended up showing that the set of all rational numbers of the form m/n, where both m and n are perfect squares (and positive), with the operation of multiplication is a subgroup of the positive rationals—I called this subgroup G. Then, I showed that the mapping that sends the rational number m/n to the element m^2/n^2 of G is an isomorphism. Here’s my full solution:

Exercise 60 is simple once we solve exercise 58. Although exercise 58 specifically discussed automorphisms of the rationals, the proof that I provided for it actually works for any arbitrary homomorphism of the rationals, so it also applies to isomorphisms of the rationals onto a proper subgroup of itself. My solution involved supposing that there was a proper subgroup of the rationals under addition that was isomorphic to the rationals under addition, and then showing that such a subgroup actually had to contain every single rational number, and thus was not a proper subgroup.

The exercise that I had the most trouble with was 47:

(The inner automorphism induced by x maps each of element a of the group to the element xax^-1.)

Here is my complete solution:

I was able to deduce fairly quickly that x and y must commute, but I had trouble going from that point to figuring out what this said about x and y as permutations.

This post is getting rather long, so I’ll leave the coset exercises that I did for the next post. (That’s the chapter that had the exercise that we still haven’t figured out how to solve.)

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