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Section5.1Preparation

ObjectivesGoals
  • Identify in context, and use "stack notation" and cycle notation to express, permutations of a finite set.
  • Simplify products and determine inverses of permutations to illuminate the structure of \(S_n\text{,}\) the group of permutations of \(n\) symbols.
  • Represent any permutation as a product of transpositions, and determine its sign (even/odd).
  • Identify how properties of the alternating subgroup \(A_n \lt S_n\) consisting only of even permutations compare to the properties of the full symmetric group \(S_n\text{.}\)
  • Discuss the reason why any finite group is "the same" as a subgroup of some symmetric group.

Subsection5.1.1Videos

Subsection5.1.2Homework 5

Homework exercises are valuable preparation for your quiz on this material. Generally I do not collect homework for grading -- however, I will expect to review your homework progress in the event of your requesting a revision on a quiz. You are welcome to discuss homework problems with your groups and with the class, both in person and in the Slack #math channel.

Subsubsection5.1.1Exercises

  1. Exercises 5.4, #1.
  2. Exercises 5.4, #2.
  3. Exercises 5.4, #3.
  4. Exercises 5.4, #5. Hint

    Since \(S_4\) is not a cyclic group (why?), listing all its subgroups will take more effort and thinking than it did for cyclic groups. There are a lot of subgroups of order 2, for example...

  5. Exercises 5.4, #8.
  6. Exercises 5.4, #10. Hint

    Begin by decomposing an arbitrary \(\sigma\) into a product of disjoint cycles; this then boils down to the question "what's the largest possible least common multiple I can create?"

  7. Exercises 5.4, #9.
  8. Exercises 5.4, #11.
  9. Exercises 5.4, #23. Hint

    This is dangerously close to Proof Portfolio problem 2.3.3.

  10. Exercises 5.4, #35. Hint

    Note that the author means "for all" \(\alpha,\beta\in S_n\text{,}\) the permutation \(\alpha^{-1}\beta^{-1}\alpha\beta\) is even. The expression

    \begin{equation*} \alpha^{-1}\beta^{-1}\alpha\beta \quad \text{is called the "commutator" of }\alpha,beta. \end{equation*}