Week 4 Problems

These problems are drawn from Chapter 5 of the text, pages 71-73.

  1. (Problem 5) Give 4 different examples of subgroups of $S_{4}$. Find each of the following sets:

    • \[\{\sigma\in S_{4} : \sigma(1) = 3\}\]
    • \[\{\sigma\in S_{4} : \sigma(2) = 2\}\]
    • \[\{\sigma\in S_{4} : \sigma(1)=3\mathrm{\ and\ }\sigma(2)=2\}\]

  2. (Problem 13) Let \(\sigma\in S_{n}\) satisfy \(\sigma=\sigma_1\sigma_2\cdots\sigma_m\) where the $\sigma_{i}$ are disjoint cycles. Prove that the order of $\sigma$ is the least common multiple of the lengths of the $\sigma_{i}$. (Hint: look at some examples first.)

  3. (Problem 23) If $\sigma\in S_{n}$ is a cycle of odd length, prove that $\sigma^2$ is also a cycle.

  4. (Problem 26) Prove that any element $\sigma$ in $S_{n}$ can be written as a finite product of the permutations in each bullet point below.

    • $(12),(13),\ldots,(1n)$
    • $(12),(23),\ldots,(n-1,n)$
    • $(12)$ and $(12\ldots n)$

  5. (Problem 27) Let $G$ be a group and let $g\in G$ be any element. Prove that the map $\lambda_{g}:G\to G$ given by $\lambda_{g}(a)=ga$ is bijective, and therefore is a permutation of $G$.

  6. (Problem 29) Recall that the center $Z(G)$ of a group $G$ is the subgroup \(Z(G) = \{g\in G: gx=xg\mathrm{\ for all\ }x\in G\}\) What is the center of $D_{n}$? Hint: look at some examples first.

  7. (Problem 31) Define a relation $\sim$ on $S_{n}$ by $a\sim b$ if and only if there exists $\sigma\in S_{n}$ so that $\sigma a \sigma^{-1} = b$. Prove that $\sim$ is an equivalence relation.

  8. (Problem 33) Prove that the center of $S_{n}$ is the trivial subgroup if and only if $n\ge 3$.