Math 360, Fall 2013, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. Direct product (of two groups).
2. Direct sum (of two abelian groups written additively).
3. Isometry (of $\mathbb{R}^n$).
4. Symmetry group (of a subset of $\mathbb{R}^n$).

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

1. Theorem relating $\mathbb{Z}_m\times\mathbb{Z}_n$ to $\mathbb{Z}_{mn}$ (Theorem 11.5 in the text).
2. Fundamental Theorem of Finitely Generated Abelian Groups (Theorem 11.12).

Solve the following problems:

1. Section 11, problems 1, 3, 8, 23, and 25.
2. Section 12, problems 1, 5, and 7.

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

Questions:

1. 11.1

   List the elements of \(\mathbb{Z}_2\times \mathbb{Z}_4\). Give the order for each element. Is the group cyclic?

2. 11.3

   Find the order of \((2,6)\) in \(\mathbb{Z}_4\times \mathbb{Z}_{12}\).

3. 11.8

   What is the largest order among the orders of all the cyclic subgroups of \(\mathbb{Z}_6\times \mathbb{Z}_8\)? of \(\mathbb{Z}_{12}\times \mathbb{Z}_{15}\)?

4. 11.23

   Find all abelian groups up to isomorphism of order 32.

5. 11.25

   Find all abelian groups up to isomorphism of order 1089.

6. 12.1

   a. Describe all symmetries of a point in the real line \(\mathbb{R}\).

   b. Describe all symmetries of a point in the plane \(\mathbb{R}^2\).

   c. Describe all symmetries of a line segment in \(\mathbb{R}\).

   d. Describe all symmetries of a line segment in \(\mathbb{R}^2\).

   e. Describe some symmetries of a line segment in \(\mathbb{R}^3\).

7. 12.5

   Draw a plane figure that has a two-element group as its group of symmetries in \(\mathbb{R}^2\).

8. 12.7

   Draw a plane figure that has a four-element group isomorphic to \(\mathbb{Z}_4\) as its group of symmetries in \(\mathbb{R}^2\).

Solutions:

Definitions:

1. Direct Product of Two Groups.

   Definition:

   Take two groups \(G\) and \(H\). Take their Cartesian product \(G\times H\). Define the binary operation \(\cdot\) on this set as:$$ (a,b)\cdot(c,d) = (ac,bd) $$

   Where \(a,c\in G\) and \(b,d\in H\). The binary structure \(G\times H, \cdot\) is the direct product of \(G\) and \(H\).

   Example:

   The direct product of \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\) is \((0,0),(0,1), (0,2), (1,0), (1,1), (1,2)\)

   Non-Example:

   \(\mathbb{Z}_4\) is not the direct product of \(\mathbb{Z}_2\) and \(\mathbb{Z}_2\).

2. Direct Sum of Two Abelian Groups

   Definition:

   A direct sum is a just a direct product of abelian groups - groups written additively.

   Example:

   \(\mathbb{Z}\times\mathbb{Z}\) is the direct sum of \(\mathbb{Z}\).

   Non-Example:

   \(GL(n,\mathbb{R})\times GL(n,\mathbb{R})\) is not a direct sum - the underlying group is not abelian.

3. Isometry of \(\mathbb{R}^n\)

   Definition:

   An isometry of \(\mathbb{R}^n\) is a permutation (bijective function) of \(\mathbb{R}^n\) that preserves distance. If \(\phi\) is an isometry, then:$$ d(\phi(x),\phi(y)) = d(x,y)$$

   Example:

   The translation \(f:\mathbb{R}^3\rightarrow \mathbb{R}^3\) given by:$$ f(x,y,z) = (x+1,y+1,z+1) $$

   is a isometry.

   Non-Example:

   The function \(f(x,y,z) = (x^3,y,z)\)

   is not an isometry.

4. Symmetry Group of a Subset of \(\mathbb{R}^n\)

   Definition:

   Given a subset \(A\subset \mathbb{R}^n\), the symmetry of \(A\) is the set of all isometries that map \(A\) to \(A\). So no point in \(A\) can be mapped outside of \(A\), and no point outside of \(A\) can be mapped inside \(A\).

   Example:

   The dihedral group is the symmetry group of a regular \(n\)-gon.

   Non-Example:

   Given a polygon, no translation is part of its symmetry group - translating a polygon does not result in the same polygon.

Theorems:

1. Theorem Relating \(\mathbb{Z}_m\times \mathbb{Z}_n\) to \(\mathbb{Z}_{mn}\)

   \(\mathbb{Z}_m\times \mathbb{Z}_n\) is isomorphic to \(\mathbb{Z}_{mn}\) if and only if \(gcd(m,n)=1\).

2. Fundamental Theorem of Finitely Generated Abelian Groups

   Take a finitely generated abelian group \(G\). This group is isomorphic to:$$ \mathbb{Z}^m \times \mathbb{Z}_{p_1^{n_1}} \times \ldots \times \mathbb{Z}_{p_k^{n_k}} $$

   where \(p_i\) is prime. \(m\) is the Betti number of the group. The other factors are unique up to ordering.

Book Solutions:

1. 11.1

2. 11.3

3. 11.8

4. 11.23

5. 11.25

6. 12.1

7. 12.5

8. 12.7