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. Prove that every nonabelian group of order $6$ is isomorphic to \(S_3\).
Hint: Here’s one possible approach. Let $G$ be a nonabelian group of order $6$. To get a homomophism from \(G\) to \(S_3\), find an action of $G$ on a set of size $3$ (and then the associated permutation representation is the desired homomorphism that you then have to verify is an isomorphism). One way to do this is to find a subgroup $H$ of index $3$ and then act on $G/H$ by left multiplication. Start by explaining why $G$ must have subgroups $P$ and $Q$ with \(\vert P\vert = 2\) and \(\vert Q\vert = 3\), and then work to show that $P$ cannot be normal in $G$ (see part 2 of the first problem from Homework 03).
Problem. Prove that if \(\vert G\vert = 280=2^3\cdot 5\cdot 7\), then \(G\) is not simple.
Hint: Assume \(G\) is simple. Perform Sylow analysis, and then try counting elements. When \(p = 5,7\), the Sylow $p$-subgroups have prime order, so as they intersect trivially, you should be able to exactly count the elements of order $p$. However, the Sylow $2$-subgroups may intersect each other nontrivially, but assuming \(n_2 > 1\), you can still say for sure that there are (strictly) more than \(7\) elements of order 2, 4, or 8. And of course there is also the identity.
Problem. Let \(p\) be prime. Prove that if \(\vert G\vert = 8p^2\), then \(G\) is not simple.
Hint: Assume \(G\) is simple. Start with Sylow analysis, and show that there are only two possibilities for the prime \(p\). Then analyze each case separately. If \(n_p\) is large, try counting elements, and if \(n_p\) is small, try acting on $G/N_G(P)$ by left multiplication (for some \(P\in \operatorname{Syl}_p(G)\)).
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 $P$ 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 $p$.
Problem. Prove that every nonabelian group of order $8$ is isomorphic to $D_8$ or $Q_8$.
Hint: Here’s a possible way to start. Let $G$ be a nonabelian group of order $8$. First, explain why \(\vert Z(G)\vert =2\), 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 \(\langle s,t\rangle\).
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.
Problems. Work some problems from D&F Section 4.5. Here are some to consider: 4, 12, 16
Hint: On #16, try counting elements.