Homework Assignment 01

Problems to submit for grading

Please organize your work, justify your steps, and write in complete sentences with correct punctuation. Once you finish, 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!

Problem 1. Let $G$ be a group with the property that for every choice of three elements of $G$, at least two of the elements commute¹. Prove that $G$ is abelian.

Problem 2. Let $n\ge 3$, and let $D_{2n}$ be the dihedral group of order $2n$ viewed as the automorphism group of the graph $\mathbb{D}_n$ that represents a regular $n$-gon with vertices labeled $1,2,\ldots,n$ clockwise. Let $r\in D_{2n}$ be the permutation of the vertices that sends vertex $i$ to $i+1$ (mod $n$), and let $s\in D_{2n}$ be the permutation sending $i$ to $-i$ (mod $n$). So $r$ corresponds to clockwise rotation by $2\pi/n$ and $s$ is a reflection. Then the following are true:

$r^n = 1$, $s^2= 1$, $srs = r^{-1}$;
$r^k \neq 1$ for all $1\le k < n$;
$D_{2n} = \{1,r,r^2,\ldots,r^{n-1}, s, sr, sr^2, \ldots, sr^{n-1}\}$.

Prove that $Z(D_{2n}) = \{1\}$ if and only if $n$ is odd. (Recall that $Z(D_{2n})$ denotes the center² of $D_{2n}$.)
Hint: first, notice that, since $D_{2n}$ is generated by $r$ and $s$, we have that $z\in Z(D_{2n})$ if and only if $z$ commutes with $r$ and $s$. Also, notice/prove that the relation $srs = r^{-1}$ implies that $sr^ks = r^{-k}$ for all integers $k$, and the latter in turn implies that $sr^k = r^{-k}s$ (using that $s$ is its own inverse).

Problem 3. Let $G$ be a finite group. Suppose there exists $\alpha\in \operatorname{Aut}(G)$³ such that

$\alpha$ is an involution⁴ (i.e. $\alpha$ is not the identity function on $G$ but $\alpha^2$ is);
$\alpha(x) \neq x$ whenever $x\neq 1$.

Prove that $G$ is abelian by following the steps below.

Let $Y \subseteq G$ be defined by $Y = \{g\in G \mid g= h^{-1}\alpha(h) \text{ for some $h\in G$}\}$. Prove that if $g\in Y$ then $\alpha(g) = g^{-1}$.
Prove that $Y= G$.
Hint: Consider the map $\gamma: G\rightarrow G: h\mapsto h^{-1}\alpha(h)$. (Note that $\gamma$ need not be a homomorphism.) Observe that proving $Y= G$ is the same as proving $\gamma$ is surjective; moreover, to show $\gamma$ is surjective, you can instead show that $\gamma$ is injective (but make sure you explain why).
The first two points combine to show that $\alpha(x) = x^{-1}$ for all $x\in G$. Use this to prove that $G$ is abelian.

Problems for extra practice

Please try to solve some of these problems, but you do not turn them in. I’m happy to talk about these problems (or any others) if you have questions.

Problem. Let $G$ be a group in which every nontrivial element is an involution. Prove that $G$ is abelian.

Problem. Let $F$ be a field, and let $\mathcal{S}=\{ A\in \operatorname{GL}_n(F) \mid \text{$A = \lambda I_n$ for some $0\neq \lambda\in F$}\}$ be the set of all nonzero, scalar $n\times n$ matrices over $F$. Prove that $Z(\operatorname{GL}_n(F)) = \mathcal{S}$.
Hint: proving that $\mathcal{S} \subseteq Z(\operatorname{GL}_n(F))$ should be fairly straightforward. To complete the proof, take an arbitrary $B\notin \mathcal{S}$, and find a matrix that does not commute with it. You don’t need to be too creative; consider various Elementary matrices.

Problems. work some problems from D&F Section 1.1. Here are some to consider: 14, 29, 31, 32, 35, 36.

Problems. work some problems from D&F Section 1.6. Here are some to consider: 13, 14, 18, 19, 20, 22.

Footnotes

Definition. For $g,h\in G$ a group, $g$ and $h$ are said to commute if $gh=hg$. ↩
Definition. For $G$ a group, the center of $G$ is defined to be $Z(G)=\{ g\in G\mid \forall h\in G(gh=hg)\}$. ↩
Definition. Let $G$ be a group. An automorphism of $G$ is an isomorphism from $G$ to $G$. Let $\operatorname{Aut}(G)$ denote the collection of all automorphisms of $G$; it’s a group called the automorphism group of $G$. ↩
Definition. An element of a group is called an involution if it has order 2. ↩