Math 360, Fall 2013, Assignment 9

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

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

1. Homomorphism.
2. Monomorphism.
3. Epimorphism.
4. Trivial homomorphism.
5. Projection homomorphism (Example 13.8 in the text).
6. Reduction modulo $n$.
7. Image (of a set under a function).
8. Pre-image (of a set under a function).
9. Fiber (of a homomorphism over a point).
10. Kernel (of a homomorphism).
11. Factor group (of a group $G$ by a normal subgroup $H$).
12. Automorphism.
13. Inner automorphism.
14. Conjugation (by an element of a group).

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

1. Theorem concerning images and pre-images of subgroups (Theorem 13.12).
2. Characterization of fibers as cosets (Theorem 13.15).
3. Characterization of monomorphisms (Corollary 13.18).
4. Fundamental theorem on homomorphisms (Theorem 14.11).
5. Criteria for normality (Theorem 14.13).

Solve the following problems:

1. Section 13, problems 1, 3, 7, 8, 17, 47, and 48.
2. Section 14, problems 1, 5, 11, 23, and 24.

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

Solutions:

Definitions:

1. Homomorphism.

   Definition:

   Given a group \(G\) and a group \(H\), a homomorphism from \(G\) to \(H\) is a function \(\phi:G\rightarrow H\) such that \(\phi(g_1g_2) = \phi(g_1)\phi(g_2)\).

   Example:

   The function \(\phi:\mathbb{Z} \rightarrow \mathbb{R}\) defined by \(\phi(z) = z\) is a homomorphism.

   Non-Example:

   The function \(phi:\mathbb{Z} \rightarrow \mathbb{R}\) defined by \(\phi(z) = z^2\) is not a homomorphism.

2. Monomorphism.

   Definition:

   A monomorphism is an injective homomorphism.

   Example:

   \(\phi:\mathbb{Z}\rightarrow \mathbb{R}\) defined by \(\phi(z) = z\) is a monomorphism.

   Non-Example:

   The function \(\phi:\mathbb{R}\rightarrow \mathbb{Z}\) defined by \(\phi(r) = \lfloor r \rfloor\) is not a monomorphism.

3. Epimorphism.

   Definition:

   An epimorphism is a surjective homomorphism.

   Example:

   The function \(\phi:\mathbb{R}\rightarrow \mathbb{Z}\) defined by \(\phi(r) = \lfloor r \rfloor\) is an epimorphism.

   Non-Example:

   \(\phi:\mathbb{Z}\rightarrow \mathbb{R}\) defined by \(\phi(z) = z\) is not an epimorphism.

4. Trivial Homomorphism.

   Definition:

   The trivial homormorphism maps \(G\) to the identity of another group \(H\).

   Example:

   Non-Example:

5. Projection Homomorphism.

   Definition:

   Let \(G = G_1\times \cdots \times G_n\). The projection homomorphism from \(G\) to \(G_i\) is given by \(\pi(g_1,g_2,\ldots,g_i,\ldots,g_n) = g_i\).

   Example:

   The homomorphism \(\pi:\mathbb{Z}_4\times \mathbb{Z}_5\rightarrow \mathbb{Z}_5\) with \(\pi(a,b) = b\) is a projection homomorphism.

   Non-Example:

6. Reduction modulo \(n\).

   Definition:

   Define \(\phi_n:\mathbb{Z}\rightarrow \mathbb{Z}_n\) by \(\phi_n(z) = z%n\) (% = modulo). This is a homomorphism.

   Example:

   Non-Example:

7. Image of a Set under a Function.

   Definition:

   Let \(f:A\rightarrow B\). Let \(C \subseteq A\). The image of \(C\) under \(f\) is the set \(f[C] = \{f(x)|x\in C\}\)

   Example:

   Non-Example:

8. Pre-Image of a Set under a Function.

   Definition:

   Let \(f:A\rightarrow B\), and \(D\subseteq B\). The pre-image of \(D\) under \(f\) is the set \(f^{-1}[D] = \{a | f(a)\in D\}\)

   Example:

   Non-Example:

9. Fiber of a Homomorphism over a Point.

   Definition:

   \(\phi:G\rightarrow H\) is a homomorphism. The fiber of \(\phi\) over \(h\in H\) is the pre-image of the singleton set \(\{h\}\), i.e. it is the set \(f^{-1}[\{h\}] = \{g | f(g) = h\}\)

   Example:

   Non-Example:

10. Kernel of a Homomorphism.

    Definition:

    The kernel of a homomorphism is the fiber of the identity.

    Example:

    Non-Example:

11. Factor Group.

    Definition:

    \(G\) a group, \(H\) a normal subgroup. The factor group \(G/H\) is the set of left cosets of \(H\), with the operation defined by \((aH)(bH) = (ab)H\).

    Example:

    Non-Example:

12. Automorphism.

    Definition:

    An automorphism is a homomorphism from a set to itself.

    Example:

    Non-Example:

13. Inner Automorphism.

    Definition:

    \(G\) is a group, \(g\) an element of \(G\). An inner automorphism by \(g\) is a function \(\phi_g:G\rightarrow G\) given by \(\phi_g(x) = gxg^{-1}\).

    Example:

    Non-Example:

14. Conjugation by an Element of a Group.

    Definition:

    \(G\) is a group, \(g\) and \(h\) are elements. Conjugation of \(h\) by \(g\) is just applying the inner automorphism by \(g\) to \(h\).

    Example:

    Non-Example: