Math 360, Fall 2013, Assignment 3

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. Isomorphism.
2. Isomorphic.
3. Structural property.
4. Identity element.
5. Group.
6. Inverse element.
7. Abelian group.

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

1. Uniqueness of identity element.
2. Left and right cancellation laws.

Solve the following problems:

1. Section 3, problems 2, 3, 4, 8, 9, 10, and 17.
2. Section 4, problems 3, 4, 5, 6, 10, 11, 12, and 13.

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

Solutions:

1. Isomorphism.

   Definition:

   An isomorphism from a binary structure \((A,*)\) to another binary structure \((B,\circ)\) is a function \(\varphi:A\rightarrow B\) such that \(\varphi\) is bijective, and:

   \(\forall x,y \in A: \varphi(x*y) = \varphi(x)\circ \varphi(y)\)

   Example:

   Take the binary structures (from class) of \((\mathbb{Z}_n,+_n)\) and \((U_n,\cdot)\). The function \(\varphi:\mathbb{Z}_n \rightarrow U_n\) given by \(\varphi([x]) = \omega^x\), where \(x\in\mathbb{Z}\) and \(\omega = \cos\frac{2\pi}{n} + i\sin\frac{2\pi}{n}\) is an isomorphism.

   Non-Example:

2. Isomorphic

   Definition:

   Example:

   Non-Example:

3. Structure Property

   Definition:

   Example:

   Non-Example:

4. Identity Element

   Definition:

   Example:

   Non-Example:

5. Group

   Definition:

   Example:

   Non-Example:

6. Inverse Element

   Definition:

   Example:

   Non-Example:

7. Abelian Group

   Definition:

   Example:

   Non-Example: