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Math 417课程简介
Math 417 is an introduction to abstract algebra. The main objects of study are groups, which are abstract mathematical objects that reflect the most basic features of many other mathematical constructions. We will also study rings and fields and other abstract mathematical objects, which can be thought of as groups with additional structure.
Abstract algebra is a branch of mathematics that studies algebraic structures, which are sets of elements with operations defined on them. These structures include groups, rings, and fields, which are some of the most important objects of study in algebra.
In abstract algebra, we focus on the properties of these algebraic structures that are independent of the specific elements they contain. This allows us to develop general theories and techniques that apply to a wide range of mathematical objects.
Groups, as you mentioned, are one of the most fundamental algebraic structures studied in abstract algebra. A group is a set of elements together with an operation that satisfies certain axioms, such as associativity and the existence of an identity element and inverses for each element. Groups have many important applications in mathematics, physics, and computer science.
Rings and fields are more complex algebraic structures that generalize the properties of groups. A ring is a set of elements with two operations, usually addition and multiplication, that satisfy certain axioms. Fields are a type of ring in which every nonzero element has a multiplicative inverse.
Abstract algebra also studies other algebraic structures, such as modules, vector spaces, and algebras, which are extensions of the basic ideas of groups, rings, and fields. These structures have important applications in diverse fields such as number theory, cryptography, and quantum mechanics.
Prerequisites
To learn Math 417, an introduction to abstract algebra, it is recommended that you have a solid understanding of some basic mathematical concepts, including:
- Set theory: A basic understanding of set theory is essential, as algebraic structures such as groups, rings, and fields are sets with additional algebraic operations defined on them.
- Logic and proofs: Abstract algebra involves rigorous proof-based reasoning, so it is important to have a good grasp of mathematical logic and the basics of proof-writing.
- Linear algebra: Linear algebra is another fundamental branch of mathematics that provides many of the tools and concepts used in abstract algebra. You should have a good understanding of vectors, matrices, linear transformations, and determinants.
- Calculus: A basic knowledge of calculus, including limits, derivatives, and integrals, is also useful in understanding certain concepts in abstract algebra.
It is important to note that the specific prerequisites for Math 417 may vary depending on the institution and instructor teaching the course. Therefore, it is always a good idea to check the course syllabus or speak with the instructor to ensure that you have the necessary background knowledge to succeed in the course.
MATH 416 or one of MATH 410, MATH 415 together with one of MATH 347, MATH 348, CS 374; or consent of instructor
MATH 417 assignment help（exam help， online tutor）
Prove that subgroup of a cyclic group is also cyclic.
To prove that any subgroup of a cyclic group is also cyclic, let $G$ be a cyclic group and let $H$ be a subgroup of $G$. Let $g$ be a generator of $G$, so that every element of $G$ can be written as a power of $g$. Since $H$ is a subgroup of $G$, it contains the identity element $e$, so we can write $e=g^0\in H$.
Let $h$ be any non-identity element of $H$. Since $H$ is a subgroup of $G$, $h$ must also be an element of $G$, so we can write $h=g^k$ for some integer $k$. Since $h$ is not the identity element, $k\neq 0$.
We claim that $H=\langle h\rangle={g^k: k\in\mathbb{Z}}$. To see this, note that if $x\in H$, then $x$ can be written as $x=g^m$ for some integer $m$, since $G$ is cyclic. Since $H$ is a subgroup of $G$, we have $g^m\in H$. If $m=0$, then $x=e\in\langle h\rangle$. If $m\neq 0$, then we can write $m=qk+r$ for some integers $q$ and $r$, where $0\leq r<|k|$. Then we have $g^m=g^{qk}g^r=(g^k)^qg^r\in\langle h\rangle$, since $g^k=h$ is in $\langle h\rangle$. Thus, every element of $H$ is in $\langle h\rangle$. Conversely, every element of $\langle h\rangle$ is in $H$, since $\langle h\rangle$ is a subgroup of $G$ and hence of $H$. Thus, we have $H=\langle h\rangle$, which means that $H$ is cyclic.
Show that $A_n$ is a normal subgroup of $S_n$ and compute $S_n / A_n$, that is, find a known group to which $S_n / A_n$ is isomorphic.
To show that $A_n$ is a normal subgroup of $S_n$, we need to show that for every $\sigma\in S_n$ and every $\tau\in A_n$, we have $\sigma\tau\sigma^{-1}\in A_n$. Let $\sigma\in S_n$ and let $\tau\in A_n$. We need to show that $\sigma\tau\sigma^{-1}\in A_n$, which means that we need to show that $\sigma\tau\sigma^{-1}$ is an even permutation.
Since $\tau$ is an even permutation, we have $\mathrm{sgn}(\tau)=1$. Thus, we have
\begin{align*} \mathrm{sgn}(\sigma\tau\sigma^{-1})&=\mathrm{sgn}(\sigma)\mathrm{sgn}(\tau)\mathrm{sgn}(\sigma^{-1})\ &=\mathrm{sgn}(\sigma)\mathrm{sgn}(\tau)\mathrm{sgn}(\sigma)\ &=(\mathrm{sgn}(\sigma))^2\mathrm{sgn}(\tau). \end{align*}
Since $\sigma$ is a permutation, we have $\mathrm{sgn}(\sigma)=\pm 1$. Thus, we have $\mathrm{sgn}(\sigma)^2=1$, which means that
$$\mathrm{sgn}(\sigma\tau\sigma^{-1})=\mathrm{sgn}(\tau).$$
Since $\tau$ is an even permutation, we have $\mathrm{sgn}(\tau)=1$, which means that $\sigma\tau\sigma^{-1}$ is also an even permutation. Thus, we have $\sigma\tau\sigma^{-1}\in A_n$, which means that $A_n$ is a normal subgroup of $S_n$.
To compute $S_n/A_n$, we need to find a known group to which $S_n/A_n$ is isomorphic. Let $N=A_n$. Then by the First Isomorphism Theorem, we have
$$S_n/N\cong \frac{S_n}{\mathrm{ker}(\varphi)},$$
where $\varphi:S_n\rightarrow S_n/N$ is the natural projection homomorphism. Note that $\mathrm{ker}(\varphi)=A_n$, since $\varphi(\sigma)=\sigma N$ is the identity element of $S_n/N$ if and only if $\sigma\in A_n$. Thus, we have
$$S_n/A_n\cong \frac{S_n}{A_n}.$$
To compute $S_n/A_n$, we need to determine the index $m=|S_n:A_n|$. Note that $|S_n|=n!$ and $|A_n|=\frac{1}{2}|S_n|=\frac{1}{2}n!$, so we have
$$m=\frac{|S_n|}{|A_n|}=2.$$
Let $N \triangleleft G$. If $m=|G: N|$ then show that $a^m \in N$ for every $a \in G$
Thus, $S_n/A_n$ is a group of order 2. The only group of order 2 is the cyclic group $\mathbb{Z}/2\mathbb{Z}$, so we have
$$S_n/A_n\cong \mathbb{Z}/2\mathbb{Z}.$$
To prove that $a^m\in N$ for every $a\in G$, let $N\triangleleft G$ and let $m=|G:N|$. For any $a\in G$, we have $aN\in G/N$, which means that $|aN|=|N|=m$. Since $aN$ is a coset of $N$, we have $aN={an: n\in N}$. Thus, the powers of $a$ in $aN$ are
$$a^0N,a^1N,a^2N,\ldots, a^{m-1}N.$$
Since $|aN|=m$, the powers of $a$ in $aN$ must repeat after $m$ terms. That is, there exist integers $0\leq r<s\leq m-1$ such that $a^rN=a^sN$. Then we have
$$a^r=a^r e=a^r (nn^{-1})=(a^s)^{-1}a^sa^r=(a^s)^{-1}a^{s+r}=a^{s-r}.$$
Thus, we have $a^{s-r}=a^r\in N$, which means that $a^m=a^{s-r}a^{m-(s-r)}\in N$. Therefore, $a^m\in N$ for every $a\in G$.
Tuition
| Undergraduate Students | $1248 |
| Graduate Students | $1500 |
| Courseware Cost | None |
Students must be able to view assignments online, write out solutions, then scan or take photos of their written work and upload it to Moodle.
Students with a Bachelor’s degree will be assessed graduate level tuition rate for this course. However, one cannot receive graduate level credit for courses numbered below 400 at the University of Illinois.
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