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:

   The function \(f\)from the binary structure \((\mathbb{Q},+)\) to \((\mathbb{R},+)\) given by \(f(x) = x\) is not an isomorphism, because it is not surjective. It does meet the other two requirements for an isomorphism.

2. Isomorphic

   Definition:

   Two binary structures are isomorphic if there is an isomorphism between them.

   Example:

   \((\mathbb{Z}_n,+_n)\) is isomorphic to \((U_n,\cdot)\).

   Non-Example:

   The structures \((\mathbb{Q},+)\) and \((\mathbb{R},+)\) are not isomorphic.

3. Structural Property

   Definition:

   A structural property is a property that is preserved under isomorphism. Let's say we have two binary structures \((A,*)\) and \((B,\circ)\). If we also have a property of binary structures, i.e. a statement about the structures that is either true or false, then this property is structural if it is preserved by the isomorphism. That is to say, if \((A,*)\) has the property, and there is an isomorphism from \((A,*)\) to \((B,\circ)\), then \((B,\circ)\) also has the property. I think second-order logic might be required to give a proper mathematical definition of this, so I'm going to leave it like this.

   Example:

   Commutativity of a binary structure's operation is structural.

   Non-Example:

   The type of the elements contained in a binary structure (real numbers, complex numbers, etc.) is not a structural property. Meaning the answer to the question "Are the elements of this binary structure elements of the set 'X'?" is not a structural property.

4. Identity Element

   Definition:

   Given an algebraic structure \((A,*)\), an identity element \(i\) is one such that:

   \(\forall a \in A: a*i = a = i*a\)

   So it's an element that acts the same way as 1 for multiplication (of complex numbers) or 0 for addition (of complex numbers).

   Example:

   Non-Example:

5. Group

   Definition:

   Example:

   Non-Example:

6. Inverse Element

   Definition:

   Example:

   Non-Example:

7. Abelian Group

   Definition:

   Example:

   Non-Example: