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:[edit]
- Homomorphism.
- Monomorphism.
- Epimorphism.
- Trivial homomorphism.
- Projection homomorphism (Example 13.8 in the text).
- Reduction modulo $n$.
- Image (of a set under a function).
- Pre-image (of a set under a function).
- Fiber (of a homomorphism over a point).
- Kernel (of a homomorphism).
- Factor group (of a group $G$ by a normal subgroup $H$).
- Automorphism.
- Inner automorphism.
- Conjugation (by an element of a group).
Carefully state the following theorems (you need not prove them):[edit]
- Theorem concerning images and pre-images of subgroups (Theorem 13.12).
- Characterization of fibers as cosets (Theorem 13.15).
- Characterization of monomorphisms (Corollary 13.18).
- Fundamental theorem on homomorphisms (Theorem 14.11).
- Criteria for normality (Theorem 14.13).
Solve the following problems:[edit]
- Section 13, problems 1, 3, 7, 8, 17, 47, and 48.
- Section 14, problems 1, 5, 11, 23, and 24.
Questions:[edit]
- With regard to the last three definitions (automorphism, inner automorphism, conjugation). I don't believe we formally introduced these terms in class. With Monday's quiz and Wednesday's exam coming up, should I just use the definitions from the Wiki or is there more that will be said in class?--Robert.Moray (talk) 11:37, 2 November 2013 (EDT)
- I saw that one of the homework questions has to do with showing that $A_n$ is a normal subgroup of $S_n$. I think we proved this in class using the first of the criteria for normality, but I wrote a proof here using the third criteria. Let me know what you think! --Robert.Moray (talk) 12:49, 3 November 2013 (EST)
Solutions:[edit]
Definitions:[edit]
- 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.
- 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.
- 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.
- Trivial Homomorphism.
Definition:
The trivial homormorphism maps \(G\) to the identity of another group \(H\).
Example:
Non-Example:
- 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:
- 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:
- 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:
- 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:
- 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:
- Kernel of a Homomorphism.
Definition:
The kernel of a homomorphism is the fiber of the identity.
Example:
Non-Example:
- 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:
- Automorphism.
Definition:
An automorphism is a homomorphism from a set to itself.
Example:
Non-Example:
- 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:
- 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:
Theorems[edit]
- Theorem Concerning Images and Pre-Images of Subgroups
The image and pre-image of a subgroup is also a subgroup.
- Characterization of Fibers as Cosets
Given \(\phi:G\rightarrow H\), and \(g\in G\), the fiber of \(\phi(g)\) is \(g (ker\phi)\)
- Characterization of Monomorphisms
A homomorphism \(\phi : G \rightarrow H\) is a monomorphism if and only if \(\ker(\phi) = \{e\}\).
- Fundamental Theorem on Homomorphisms
Let \(\phi:G\rightarrow H\)be a homomorphism. There is an epimorphism from \(G\) to \(G/ker\phi)\). Call it \(\pi\). There is also an isomorphism \(\psi:G/ker\phi\rightarrow \phi[G]\) given by \(\psi(gK) = \phi(g)\). \(\psi\) is unique, and can also be thought of as a monomorphism into \(H\).
- Criteria for Normality
A subgroup \(H\) is normal if and only if \(H\) is the kernel of some homomorphism \(\phi\).
