## Math 417 Homework 8: Due Friday March 28

*Instructions*: You can discuss these problems with others,
but write up your solutions on your own (i.e., don't just
copy someone else's solutions, else the feedback I give
you won't help you much). Please be neat and write in full sentences.

Do four of the five problems.

- [1] Problem #30 on p. 131.

- [2] Problem #38 on p. 132.

- [3] Problem #10 on p. 145.

- [4] Problem #12 on p. 145.

- [5] Let G be the set of all infinite sequences of integers (a
_{1}, a_{2}, ... )
which have only finitely many nonzero entries. (Thus every element of G is a sequence
whose entries are integers, and those integers are all eventually 0.)
- Prove that G is a group under componentwise addition. (I.e.,
if
**x** = (x_{1}, x_{2}, ... ) and **y** = (y_{1}, y_{2}, ... )
are in G, then **x** + **y** = (x_{1} + y_{1}, x_{2} + y_{2}, ... ).)
- Let
**Q**^{+} be the group of positive rational numbers under ordinary
multiplication. Prove that G is isomorphic to **Q**^{+}.