Math 360, Fall 2016, Assignment 4

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

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

1. Semigroup.
2. Monoid.
3. Group.
4. Abelian group.
5. Inverse element.
6. Order (of a group; look in your textbook for this one).
7. $M_n(\mathbb{R})$.
8. $GL_n(\mathbb{R})$.
9. $\mathrm{Sym}(S)$.
10. Permutation (of a set $S$).
11. $S_n$.
12. Isometry (of $\mathbb{R}^n$).
13. $\mathrm{Iso}(\mathbb{R}^n)$.
14. Multiplicative notation.
15. Additive notation.
16. Power (of an element of a group written multiplicatively).
17. Multiple (of an element of an abelian group written additively).

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

1. Theorem concerning uniqueness of inverses.
2. Existence and uniqueness of solutions of equations of the forms $a*x=b$ and $x*a=b$.
3. Theorem concerning the order of $S_n$.
4. Laws of exponents (in a group, written in multiplicative notation).
5. Laws of multiples (in an abelian group, written in additive notation).

Solve the following problems:

1. Give an example of each of the following: (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.
2. Prove that the composition of two isometries of $\mathbb{R}^n$ is an isometry of $\mathbb{R}^n$.
3. Section 3, problems 31 and 32.
4. Section 4, problems 1, 3, 5, 10, 11, 12, and 17.

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