Math 360, Fall 2019, Assignment 5

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

Read:

1. Section 3.
2. Section 4.

Carefully define the following terms, then give one example and one non-example of each:

1. Isomorphism (from one binary structure to another).
2. Isomorphic (binary structures).
3. Structural property.
4. Semigroup.
5. Monoid.
6. Group.
7. Abelian group.
8. Unit (in a monoid).
9. Group of units (of a monoid).
10. $GL_n(\mathbb{R})$ (the general linear group).
11. $\mathrm{Sym}(S)$ (the symmetric group on the set $S$).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning the uniqueness of identity elements.
2. Formula for the inverse of a product.

Solve the following problems:

1. Section 3, problems 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, and 30.
2. Section 4, problems 1, 3, 5, 7, 8, 11, 13, 15, 17, and 19.
3. Give examples of (a) a binary structure which is not a semigroup; (b) a semigroup which is not a monoid; (c) a monoid which is not a group; (d) a group which is not abelian; and (e) an abelian group.
4. Consider the group of units $U=\mathcal{U}(\mathbb{Z}_3,\cdot_3)$. (a) Write an operation table for $U$. (b) Find an isomorphism between $U$ and some more "familiar" binary structure that we've discussed in class.
5. Repeat the above exercise with $U=\mathcal{U}(\mathbb{Z}_5,\cdot_5)$.
6. Repeat the above exercise with $U=\mathcal{U}(\mathbb{Z}_p,\cdot_p)$ where $p$ is a prime of your choice. Try it for some non-primes too.

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