2.2.07
April 21, 2019
Let \(n \in \mathbb{Z}\) with \(n \ge 3\). Prove that \(Z(D_{2n}) = 1\) if \(n\) is odd and that \(Z(D_{2n}) = \{1,r^k\}\) if \(n = 2k\).
2.2.10
April 21, 2019
Let \(H\) be a subgroup of order \(2\) in \(G\). Show that \(N_G(H) = C_G(H)\). Deduce that if \(N_G(H) = G\) then \(H \le Z(G)\).
2.3.17
April 27, 2019
Find a presentation for \(Z_n\) with one generator.
2.3.25
April 27, 2019
Let \(G\) be a cyclic group of order \(n\) and let \(k\) be an integer relatively prime to \(n\). Prove that the map \(x \mapsto x^k\) is surjective. Use Lagrange's Theorem to prove the same is true for any finite group of order \(n\).