Math 360, Fall 2013, Assignment 5

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

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

1. Subgroup generated by a subset.
2. Finitely generated group (hint to produce a non-example: Theorem 7.6 implies that finitely generated groups must be countable).
3. Permutation (of a set \(A\)).
4. Symmetric group (of a set \(A\)).
5. Symmetric group (on \(n\) letters).
6. Group of permutations (be careful -- this is not a synonym for "symmetric group").
7. Dihedral group.

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

1. Cayley's Theorem.

Solve the following problems:

1. Section 7, problems 3 and 6.
2. Section 8, problems 1, 5, 7, 9, 40, 42, and 46.

--------------------End of assignment--------------------

Solutions:

Definitions

1. Subgroup Generated by a Subset.

   Definition:

   Example:

   Non-Example:

2. Finitely Generated Group.

   Definition:

   Example:

   Non-Example:

3. Permutation of a Set A.

   Definition:

   Example:

   Non-Example:

4. Symmetric Group of a Set A.

   Definition:

   Example:

   Non-Example:

5. Symmetric Group on n Letters.

   Definition:

   Example:

   Non-Example:

6. Group of Permutations.

   Definition:

   Example:

   Non-Example:

7. Dihedral Group.

   Definition:

   Example:

   Non-Example:

Theorems

1. Cayley's Theorem