Homework Assignment 02

Results to know

Please read over these problems. You can use these results in future problems even though you are not being asked to prove them.

• Section 1.3: 10, 11
• Section 1.4: 8, 11
• Section 1.6: 1, 2, 10, 16, 22, 25, 26

---

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 1.3: 2, 8, 13, 14, 18
  • Problems like #13 and #14 are meant as exploration to lead you to #15. As such, you can prove them without using #15, but alternatively, it is also valuable to prove them with the help of #15 to get practice using #15 (which is an important one to remember).
• Section 1.5: 1, 3
• Section 1.6: 13, 14, 17, 18, 19

Additional Problem. Let $F$ be a field, and let $\mathcal{S}=\{ C\in \operatorname{GL}_n(F) \mid \text{$C = \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 $Z(\operatorname{GL}_n(F)) = \mathcal{S}$.

Comments:

• Recall that the center of a group $G$, denote $Z(G)$, is $Z(G) = \{ z\in G\mid \text{$\forall g\in G$, $zg=gz$} \} $.
• Elementary matrices can be helpful here. They are usually introduced in a linear algebra class. Let $E_{ij}$ be the matrix with all 1’s on the main diagonal, a 1 in the $(i,j)$-entry, and 0’s everywhere else. Here’s what might be useful for this problem.
  • If $A$ is any matrix, then $E_{ij}A$ equals the matrix obtained by applying the row operation $r_i + r_j \rightarrow r_i$ to $A$.
  • If $A$ is any matrix, then $AE_{ij}$ equals the matrix obtained by applying the column operation $c_i + c_j \rightarrow c_j$ to $A$.

---

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 Homework 02 to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Section 1.3: 15
2. Section 1.6: 4
3. Additional Problem. Let $\alpha$ be an automorphism of a finite group $G$ such that the following hold:
   1. $\alpha$ is involutory (which means that $\alpha^2$ is the identity automorphism);
   2. $\alpha$ fixes no nontrivial element of $G$ (which means that for all $g\in G$, $\alpha(g) = g \iff g = 1$).

   Prove that $\alpha(g) = g^{-1}$ for all $g\in G$, and $G$ is abelian.

   Possible approach: (1) show that every element of the form $g= h^{-1}\alpha(h)$ for $h\in G$ has the property that $\alpha(g) = g^{-1}$; (2) show that every element of $G$ is indeed of the form $g= h^{-1}\alpha(h)$ for some $h\in G$ by showing that the map $G\rightarrow G: h\mapsto h^{-1}\alpha(h)$ is injective (and thinking about how this even helps ); (3) put these together to show that $\alpha(g) = g^{-1}$ for all $g\in G$ and use this to show $G$ is abelian.

   Concise restatement. Here’s a concise way to remember this problem: “If $\alpha$ is an involutory automorphism of a finite group $G$ with no nontrivial fixed points, then $G$ is abelian and is inverted by $\alpha$.”