Problem Set 1: Warm Up

In this problem we review doing calculations in symmetric groups. You do not need to show any work, just the final answers.

1. Below are compositions of permutations expressed in cycle notation. Express each permutation as a collection of disjoint cycles; that is, calculate the cycle notation for the composite permutation.

   1. $(134)(241)(1234)$
   2. $(15)(14)(13)(12)$
   3. $(15342)(43512)$

2. Calculate the order of the following permutations. Beware, I may be trying to trick you!

   1. $(123)(45)(67)$
   2. $(325)(146)(27)$

3. For each of the following permutations $\sigma$, determine the inverse permutation $\sigma^{-1}$.

   1. $(12345)$
   2. $(123)(456)$
   3. $(12)(345)(6789)$

   Can you see a general recipe for finding the inverse of a permutation expressed in cycle notation?

1. Give an example of an abelian group that is not cyclic (and prove it).
2. Find two non-isomorphic groups of order 6 and prove they are not isomorphic.
3. What are the possible orders of elements in the symmetric group $S_5$? For each possible order, find an element in $S_5$ with that order.
4. If $H \subseteq G$ is a subgroup with index 2, prove that $H$ is a normal subgroup of $G$.
5. Let $D_6$ be the 6th dihedral group (with order 12). Determine all the conjugacy classes of elements of $D_6$.

The table below gives the partial description of a mystery group $G$ of order 8. \[ \begin{array}{|c||c | c | c | c | c | c | c | c |} \hline G & 1 & a & b & c & d & e & f & g \\ \hline \hline 1 & & & & & & & & \\ \hline a & & \keep{\blue{1}} & & & & \keep{\blue{d}} & & \\ \hline b & & & \keep{\blue{a}} & & \keep{\blue{f}} & & & \\ \hline c & & \keep{\blue{b}} & & & & & & \\ \hline d & & & \keep{\blue{g}} & & & & & \\ \hline e & & \keep{\blue{d}} & & & & & & \\ \hline f & & & & & & & & \\ \hline g & & & & & & & & \keep{\blue{a}} \\ \hline \end{array} \]

1. Using the axioms of a group and information provided, fill in the missing entries of the multiplication table for $G$. There is only one solution!
2. Is $G$ isomorphic to $D_4$? Either find an isomorphism or prove that no such isomorphism exists.

Let $G$ be a group and suppose that $H_1, H_2 \subseteq G$ are subgroups such that $H_i \neq G$.

1. Prove that $G \neq H_1 \cup H_2$.
2. Find an example of a group $G$ which can be expressed as a union of three proper subgroups $G = H_1 \cup H_2 \cup H_3$.
3. (Challenge) Let $G$ be a group. Prove that there are three proper subgroups $H_1, H_2, H_3 \subseteq G$ such that $G = H_1 \cup H_2 \cup H_3$ if and only if there is a surjective homomorphism $\rho : G \to \ZZ/2\ZZ \times \ZZ/2\ZZ$.

Let $p$ be an odd prime and suppose that $G$ is a finite group of order $2p$. Prove that $G$ is either cyclic or $G \cong D_p$, the $p$th dihedral group.