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--------------------

Questions:

Book Problems

1. 7.3

2. 9.1

3. 9.5

4. 9.7

5. 9.9

6. 9.40

7. 9.42

8. 9.46

Solutions:

Definitions

1. Subgroup Generated by a Subset.

   Definition:

   Given a group \(G\), and a subset \(S\) of \(G\), the subgroup generated by \(S\) is the smallest subgroup containing \(S\).

   Example:

   In the integers, the subgroup generated by \(\{2,6\}\) is \(2\mathbb{Z}\) - the integer multiples of 2. (For the integers, the subgroup generated by a set is the cyclic subgroup generated by the gcd of all the elements of the set).

   Non-Example:

   The subgroup generated by \(\{4,8\}\) is not \(2\mathbb{Z}\) - while this subgroup does contain both elements, it is not the smallest subgroup that does so (that would be \(4\mathbb{Z}\).

2. Finitely Generated Group.

   Definition:

   A group \(G\) is finitely generated if there is a finite subset \(S\subseteq G\) such that the subgroup generated by \(S\) is all of \(G\).

   Example:

   The integers are finitely generated (by 1).

   Non-Example:

   Uncountable sets are not finitely generated - \(\mathbb{R}\), for instance. Any element in \(S\) can be written as \(a_1^{k_1}a_2^{k_2}\cdots a_n^{k_n}|n\in \mathbb{Z}, a_i \in S\). The number of such representations is \(\aleph_0\).

3. Permutation of a Set A.

   Definition:

   A permutation of a \(A\) is a function \(\phi:A\rightarrow A\) that is one to one and onto (bijective).

   Example:

   \(f(x)=x+1,f:\mathbb{R}\rightarrow \mathbb{R}\) is a permutation of \(\mathbb{R}\).

   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

Book Problems

1. 7.3

2. 9.1

3. 9.5

4. 9.7

5. 9.9

6. 9.40

7. 9.42

8. 9.46