Math 360, Fall 2015, 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. Structural property.
2. Semigroup.
3. Monoid.
4. Group.
5. Abelian group.
6. Inverse element.
7. Order (of a group).
8. Klein four-group.
9. Multiplicative notation.
10. Additive notation.
11. Power (of an element of a group written multiplicatively).
12. Multiple (of an element of a group written additively).
13. Induced operation (on a closed subset of a binary structure).
14. Substructure.

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

1. Existence and uniqueness of solutions of equations of the forms $a*x=b$ and $x*a=b$.
2. Theorem concerning the number of appearances of a group element in each row and each column of its group table.
3. Classification of groups of orders two and three.

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. Section 3, problems 31 and 32.
3. Section 4, problems 1, 3, 5, 7, 10, 11, 12, and 17.

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