What This Document Is
This document contains completed solutions to Homework 12 for Fordham University’s Abstract Algebra I (MATH 3005) course. The assignment covers sections 7.5 and 8.1 of the course material and was due on December 13th. It presents detailed answers to a series of problems related to permutation groups and group theory.
Why This Document Matters
This completed homework is intended for students enrolled in MATH 3005. It serves as a check for understanding of the concepts covered in sections 7.5 and 8.1, providing worked examples that can be used for self-assessment and review. It’s valuable for students seeking to verify their own solutions or understand correct approaches to the assigned problems.
Common Limitations or Challenges
This document provides *solutions* to the homework, but does not offer explanations of the underlying concepts. Students should use this as a supplement to their own work and understanding of the material, not as a replacement for learning the concepts themselves. Simply copying the solutions will not guarantee success on future assessments.
What This Document Provides
The full document includes:
* Solutions to problems involving computing products of permutations (a, s, af).
* Representations of permutations as disjoint cycles and 2-cycles.
* Calculations of the order of permutations (|a| and |β|).
* Solutions to problems involving composition of permutations (αβ, βα, α⁻¹).
* Answers to questions about the maximum order of an element in S₄.
* Proofs related to even and odd permutations in subgroups of Sₙ.
* Solutions to problems involving cosets of subgroups.
* Results related to subgroups of GL₂(ℝ) and SL₂(ℝ).
This preview does *not* include the full solutions or proofs contained within the document. It is a descriptive overview only.
