MY-ASSIGNMENTEXPERT™可以为您提供 iastate.edu MAT301 Abstract Algebra抽象代数的代写代考和辅导服务！
MAT301课程简介
Prereq: MATH 166 or MATH 166H, MATH 317, and grade of C- or better in MATH 201
Basic properties of integers, divisibility and unique factorization. Polynomial rings over a field. Congruence. Introduction to abstract rings, homomorphisms, ideals. Roots and irreducibility of polynomials. Introduction to groups. Emphasis on proofs.
Prerequisites
Math 166 (Calculus II), Math 317 or 407 (Linear Algebra), and Math 201 (Introduction to Proofs)
A student who has taken Math 207 in lieu of 317 may be prepared for the course. Discuss with the instructor. While the construction of sound proofs will be a central component of the course, a student with no previous experience writing simple proofs may find the course overly challenging.
Prerequisites
Learning Outcomes
Upon completion of this course, students…
Will be familiar with properties of the integers such as prime factorization, divisibility, and congruence
will be able to reason abstractly about mathematical structures
will recognize and comprehend correct proofs of formal statements and be able to formulate proofs clearly and concisely
Learning Objectives
Students will be able to perform computations involving divisibility of integers.
Students will be asked to identify ring-theoretic and group-theoretic properties and identify these properties in familiar rings and groups.
Students will provide proofs to simple assertions of ring- and group-theoretic principles.
MAT301 Abstract Algebra HELP（EXAM HELP， ONLINE TUTOR）
(a) Label the vertices of the hexagon with the numbers $0,1, \ldots, 5$. Explain why $D_6$ is isomorphic to a subgroup of $S_6$.
(a) We can think of the group $D_6$ as the group of symmetries of a regular hexagon. Each symmetry of the hexagon can be represented by a permutation of the vertices $0, 1, \ldots, 5$. This gives us a natural homomorphism from $D_6$ to $S_6$, since each symmetry can be thought of as permuting the vertices. This homomorphism is injective, since distinct symmetries of the hexagon give rise to distinct permutations of the vertices. Therefore, $D_6$ is isomorphic to a subgroup of $S_6$.
(b) Use Lagrange’s Theorem to find the number of cosets of $D_6$ in $S_6$.
(b) By Lagrange’s Theorem, the order of a subgroup of a finite group divides the order of the group. In this case, $|S_6| = 720$ and $|D_6| = 12$, so the number of cosets of $D_6$ in $S_6$ is $|S_6|/|D_6| = 60$.
(c) Consider the subgroup $K=\operatorname{Stab}_{S_6}(1)$ of $S_6$. Determine the order of $K$.
(c) The stabilizer of $1$ in $S_6$ is the subgroup of permutations that fix $1$ and permute the remaining five vertices. There are $5!$ permutations of the remaining five vertices, so the order of $K$ is $5! = 120$.
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