抽象代数Abstract Algebra assignment更多的经典案例请参阅以往案例,关于abstract algebra的更多的以往案例可以参阅相关文章。abstract algebra代写请认准UpriviateTA.

抽象代数就是一门概念繁杂的学科,我们最重要的一点我想并不是掌握多少例子。即便是数学工作者也不会刻意记住Jacobson环、正则环这类东西,重要的是你要知道这门学科的基本工具和基本手法,对概念理解了没有,而这一点不需要用例子来验证,只需要看看你的理解和后续概念是否相容即可

Williams大学的一次MATH 355 Commutative Rings抽象代数作业

Proof Questions


Your answers to these questions should be proofs. Problems labeled “DF” are from the textbook, Dummit and Foote.

Problem 1.

  1. Let $G$ be a group and let $H \leq G$ be a subgroup. Let $P \in S y l_p(H)$ for some prime $p$. Prove that if $P \unlhd H \unlhd G$, then $P \unlhd G$. (Recall that this statement may be false if $P \notin S y l_p(H)$ !)

Problem 2.

Let $G$ be a group of order 36 . Prove that $G$ is not simple.

Problem 3.

Let $G$ be a group. Let $|G|=p^2 q^2$, where $p<q$ are distinct primes. Prove that $G$ is not simple. (Hint: Reduce to the case where $p=2, q=3$ ).

Problem 4.

Let $G$ be a group. Let $|G|=p q r$, where $p<q<r$ are distinct primes. Prove that $G$ is not simple.

Problem 5.

Classify the simple groups of order 2 through 59 . That is, make a list all of the simple groups whose order is between 2 and 59 , inclusive. Then prove that your list is complete.
Hint: The simple groups are just the obvious ones. To prove that your list is complete, you will need to go case by case; that is, show that $Z_2$ is the only simple group of order $2, Z_3$ is the only simple group of order 3 , there are no simple groups of order 4 , etc. While this seems daunting, the vast majority of cases are covered by theorems we have proved in class or on the homework! The only cases that will require any work are 24,40 , and 56 .

Problem 6.

Classify the simple groups of order 61 through 100 .
Hint: The only cases that require work are $72,80,84$, and 90 , and the only really tricky one is 90. For 90, consider the Sylow 3-subgroups. If $n_3 \neq 1$, break the problem into 2 cases.
Case 1: For any distinct $P, Q \in \operatorname{Syl}_3(G), P \cap Q={e}$. This case can be ruled out with a simple counting argument.
Case 2: There exist distinct $P, Q \in S y l_3(G)$ such that $P \cap Q \neq{e}$. In this case, let $H=$ $P \cap Q$. Show that $N_G(H)$ has at least 18 elements, so that $\left[G: N_G(H)\right] \leq 5$.

Problem 7.

Let $G$ be a group of order $p^2$, where $p$ is a prime.
(a) Prove $G$ is abelian. (Hint: We have proved some useful facts about $p$-groups.)
(b) Prove that $G \cong Z_{p^2}$ or $G \cong Z_p \times Z_p$.

Let $G$ be a group of order $p q$, where $p<q$ are primes and $p \nmid q-1$. Prove that $G \cong Z_{p q}$. (Hint: Recall that $Z_{p q} \cong Z_p \times Z_q$.)

Problem 8.

Prove that there is no injective homomorphism from $D_{50}$ to $S_{10}$. (Hint: What does a Sylow 5 -subgroup of $S_{10}$ look like?)


Computational Questions
You don’t need to prove the answers to these questions, but you should show your work to the extent feasible.

Problem 9.

Construct a nonabelian group of order 55. After you have constructed the group, write down a presentation for it! (Hint: The techniques you used in Problem 10 of Homework 8 will be helpful.)

\begin{prob}

抽象代数代写请认准UpriviateTA

abstract algebra代写请认准UpriviateTA. UpriviateTA为您的留学生涯保驾护航。

更多内容请参阅另外一份Galois代写.

线性代数代写

Math 152 lab

math2030代写