Homework Assignment 11

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 5.1: 1, 5, 14
• Section 5.2: 1(a,b,c), 2(a,b,c), 3(a,b,c), 4(a), 7, 9
• Section 7.1: 7, 8, 11, 13, 26
• Section 7.2: 1, 7
  • For #7, take a look additional problem to try from Homework 02.

Additional Problem. Let $n\ge 1$. Suppose $H$ is a group, and $\varphi:Z_n\rightarrow \operatorname{Aut}(H): a \mapsto \varphi_a$ and $\psi:Z_n\rightarrow \operatorname{Aut}(H): a \mapsto \psi_a$ are two homomorphisms. Prove that if the images of $\varphi$ and $\psi$ are equal in $\operatorname{Aut}(H)$, then $H\rtimes_{\varphi} Z_n \cong H\rtimes_{\psi} Z_n$.

Let $Z_n = \langle x \rangle$. Using that $\varphi(Z_n) = \psi(Z_n)$, show $\varphi_x = \psi_{x^m}$ for some $m$ with $\gcd(m,n) = 1$, which implies $\varphi_a = \psi_{a^m}$ for all $a\in Z_n$. Now consider the map $\alpha: H\rtimes_{\varphi} Z_n \rightarrow H\rtimes_{\psi} Z_n : (h,a) \mapsto (h,a^m)$.

It’s worth noting that this result is true more generally in that you get the same conclusion given that $\varphi(Z_n)$ and $\psi(Z_n)$ are conjugate subgroups in $\operatorname{Aut}(H)$.

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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 there is a unique group (up to isomorphism) of order 805.

   The conclusion implies that $G$ must be abelian, so this is what you’ll want to prove first. A good place to start is by trying to show that all of the Sylow subgroups are normal. Let $P$ be a Sylow $5$-subgroup, $Q$ a Sylow $7$-subgroup, and $R$ a Sylow $23$-subgroup. Showing $Q$ and $R$ are normal should be fairly straightforward, but $P$ may be harder to control. Here’s a possible way to more forward (but there are others). First, try to show that in fact $Q,R \le Z(G)$; to do this, try following an approach similar to what you did for Exercise #24 in Section 4.5 (from Homework 10). This implies that $QR \le Z(G)$…and now what can you say about $G/Z(G)$?

2. Additional Problem. Prove that there are exactly two groups of order 21 (up to isomorphism).

   Consider showing that such a group may be represented as a semidirect product and using the Additional Problem about semidirect products to show that there is a unique nonabelian such group.

3. Additional Problem. Let $p < q$ be odd primes, and let $G$ be a group of order $p^2q^2$. Suppose that $p$ does not divide $q-1$ nor $q+1$. Prove that $G$ is abelian, and classify the possibilities for $G$ (up to isomorphism).

   The $p^2q$ example on page 144 of Dummit & Foote (and also from class) may provide some ideas.

4. Section 7.1: 14

   On part (c), it may help to think about factoring $1-x^m$.