Homework Assignment 09

Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

• Section 4.3: 19, 29, 31, 32
• Section 4.5: 1, 4, 8

Additional Problem. Let $P$ be a group of order $p^k$ for some prime $p$. If $P$ acts on a finite set $X$, then $|\operatorname{Fix}(P)| \equiv |X|$ modulo $p$.

Additional Problem. Prove that every nonabelian group of order $8$ is isomorphic to $D_8$ or $Q_8$.

Here’s a possible way to start. Let $G$ be a nonabelian group of order $8$. First, explain why $|Z(G)|=2$, so $Z(G) = \langle z \rangle$ for some central involution $z$. Now consider two cases. Case 1: $z$ is the only involution of $G$; show that this implies all other nontrivial elements of $G$ have order 4, which leads to $Q_8$. Case 2: $G$ contains some involution $s$ different than $z$; show that this implies there is yet another involution $t \neq s$ and consider $\langle s,t\rangle$.

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Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Additional Problem. Prove that every nonabelian group of order $6$ is isomorphic to $\operatorname{Sym}(3)$.

   There are various approaches, but here’s one. Let $G$ be a nonabelian group of order $6$. We want to construct an isomorphism with $\operatorname{Sym}(3)$, and a good way to get a homomophism to $\operatorname{Sym}(3)$ is to find an action of $G$ on a set of size $3$ (and then consider the associated permutation representation). We know that one way to do this is to find a subgroup $H$ of index $3$ and then act on $G/H$ by left multiplication. But, to actually get an isomorphism, you will then have to determine the kernel of the action.

   Try starting by explaining why $G$ must have subgroups $P$ and $Q$ with $|P| = 2$ and $|Q| = 3$, and then work to show that $P$ cannot be normal in $G$ (see Exercise 42 in Section 3.1).

2. Additional Problem. Let $p$ be a prime. Suppose that $P$ and $Q$ are both $p$-subgroups¹ of a group $G$. Prove that if $Q \le N_G(P)$, then $QP$ is a also a p-subgroup of $N_G(P)$.

   Please use our definition of a $p$-group (included below), and do not assume $P$, $Q$, or $G$ are finite. Consider first showing that $QP/P$ is a $p$-group.

3. Additional Problem. Let $G$ be a group of order $mp^k$ with $p$ prime and $1< m <p$. Prove that $G$ has a normal Sylow $p$-subgroup, and conclude that $G$ is not simple.

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Footnotes