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. Let \(G\) be a group, and let \(H,K\le G\).
- Prove that if \(K\) is abelian and \(K \trianglelefteq G\), then \(\langle H,K \rangle \le N_G(H\cap K)\).
- Prove that if \(H,K \trianglelefteq G\) and \(H\cap K = 1\), then \(H\le C_G(K)\).
Hint: the two parts are unrelated. For the second part, show that for all \(h\in H\) and all \(k\in K\), \(h^{-1}k^{-1}hk \in H\cap K\).
Problem. Let \(G\) be a finite group, and let \(N\trianglelefteq G\). Prove that, for all \(g\in G\), if \(\vert g\vert\) and \(\vert G:N \vert\) are relatively prime, then \(g\in N\).
Hint: how does the order of \(gN\) in \(G/N\) relate to the order of \(g\) in \(G\)? And how does the order of \(gN\) relate to \(\vert G:N \vert\)?
Problem. Let \(G\) be a group, and let \(N\le Z(G)\). Prove that if \(G/N\) is cyclic, then \(G\) is abelian.
Hint: start by choosing a generator for \(G/N\), say \(G/N = \langle aN \rangle\). Then taking arbitrary \(g,h\in G\), you can write \(gN = a^kN\) and \(hN = a^\ell N\) for some \(k,\ell \in \mathbb{Z}\). Now use this to write each of \(g\) and \(h\) in terms of powers of \(a\) and elements of \(N\). Then keep going…
Problem. Let \(G\) be a group of order \(pq\) for primes \(p\) and \(q\) (that may or may not be distinct). Prove that either \(G\) is abelian or \(Z(G)=1\).
Remark. Using the previous result as a starting point, we will later see that groups of order \(p^2\) are always abelian (for \(p\) a prime).
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.
Problems. work some problems from D&F Section 3.1. Here are some to consider: 4, 5, 14, 17, 22, 23, 29
Problems. work some problems from D&F Section 3.2. Here are some to consider: 8, 11, 14