Homework Assignment 05

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 3.1: 5, 9, 14, 21, 22, 32
Section 3.2: 4, 14
- For #14, try using the following Additional Problem to help show that $S_4$ has no normal subgroup of order $8$.

Additional Problem. Let $G$ be a finite group with a normal subgroup $N$. Set $\overline{G} = G/N$. Prove that for all $g\in G$, $|\bar{g}|$ is a common divisor of $|g|$ and $|G:N|$. Conclude that for all $g\in G$, if $|g|$ and $|G:N|$ are relatively prime, then $g\in N$.

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!

We need the following definitions.

Definition. For $G$ a group and $x,y \in G$, we define $[x,y] = x^{-1}y^{-1}xy$; it is called the commutator of $x$ and $y$.

Definition. A subgroup $H$ of a group $G$ is called central if $H \le Z(G)$.

Remark: note that every central subgroup is also normal.

Additional Problem (cf. Exercise 40 in Section 3.1). Suppose $G$ is a group with a normal subgroup $N$. Let $\bar{x},\bar{y} \in \overline{G} = G/N$. Prove that $\bar{x}$ and $\bar{y}$ commute in $\overline{G}$ if and only if $[x,y] \in N$.
Please do not use Exercise 40 from Section 3.1 in your proof.
Additional Problem (cf. Exercise 36 in Section 3.1). Let $N$ be a central subgroup of $G$. Prove that if $G/N$ is cyclic, then $G$ is abelian.
Please do not use Exercise 36 from Section 3.1 in your proof.

Possible approach: Suppose $xN$ is a generator for $G/N$ (in bar notation: $\overline{G} = \langle \bar{x} \rangle$). First show that for all $g\in G$ there exists some $n\in N$ such that $g = x^kn$. Then use this to show that $gh=hg$ for all $g,h\in G$.
Additional Problem. Let $G$ be a finite group and $p$ a prime. Suppose $N$ is a central subgroup of $G$. Prove that if $G/N$ has an element of order $p$, then $G$ has an element of order $p$.
Please do not use Cauchy’s Theorem.

Possible approach: First notice that if $N$ has an element of order $p$, then so does $G$, so in this case, we are done. Thus, it only remains to consider when $N$ does not have an element of order $p$, and in this case, $N$ is $p$-divisible by an additional problem from Homework 03. Since $\overline{G} = G/N$ has an element of order $p$, there exists some $\bar{x} \in\overline{G}$ such that $|\bar{x}| = p$. Now investigate the element $x^p$.

Remark added after the due date: Try just assuming that $N$ is normal in $G$ (but not necessarily central). Also, here is a more direct approach than what’s suggested above: let $\bar{x} \in\overline{G}$ have order $p$…what can you deduce about $|x|$?…can you find an element of order $p$ in $\langle x \rangle$?