Math 360, Fall 2020, Assignment 4

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

Read:

1. Section 4.

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

1. Structural property.
2. Semigroup.
3. Monoid.
4. Inverse (of an element of a monoid).
5. Group.
6. Abelian (group).
7. Unit (in a monoid).
8. $\mathcal{U}(M)$ (the group of units of a monoid $M$).
9. Generic, multiplicative, and additive notation.

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

1. Theorem concerning solutions of equations of the forms $a\triangle x=b$ and $x\triangle a=b$, in groups.
2. Theorem concerning the appearance of group tables.
3. Theorem characterizing groups with two elements.
4. Theorem characterizing groups with three elements.
5. Laws of exponents (written in multiplicative notation).
6. Laws of exponents (written in additive notation).

Solve the following problems:

1. Section 4, problems 1, 3, 5, 10, 11, 12, 13, 14, 18, 19, 28, and 35. (Hint for 35: write everything out explicitly, without exponents, then start cancelling what can be cancelled.)
2. Give an example 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, and (d) a group which is not abelian.
3. Show that for any set $S$, the binary structure $(\mathrm{Fun}(S,S),\circ)$ is a monoid.
4. In class, we constructed the operation table for $(\mathrm{Fun}(S,S),\circ)$ when $S$ is a set with two elements. By examining this table, determine which elements of this structure are units.
5. (Challenge) For a general set $S$, which elements of $(\mathrm{Fun}(S,S),\circ)$ are units? (The group of units $\mathcal{U}(\mathrm{Fun}(S,S))$ is extremely important in applications, and is called $\mathrm{Sym}(S)$, the symmetric group on $S$. We shall study it in great detail later in the course.)

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Solutions:

Structural property. Semigroup. Monoid. Inverse (of an element of a monoid). Group. Abelian (group). Unit (in a monoid). U(M) (the group of units of a monoid M). Generic, multiplicative, and additive notation.