Homework 17

Read: Finish Section 4.1. Read Section 4.2 and start 4.3.

• Please read Section 4.2 about Dihedral groups, even though no problems are assigned from there.

Turn in: 4.43, 4.46, 4.49, 4.53, 4.71, 4.72, 4.75

• It should be emphasized that 4.41 is a very important result about cyclic groups. It’s definitely worth memorizing, even though you’re not being asked to prove it.

• Your proof of 4.43 might be quite short; you want to make use of Theorem 4.41.

• Both 4.44 and 4.45 can be quite useful (as can Corollary 4.47). They may not be something to memorize, but you definitely want to remember that they are there. In words, 4.44 tells you the order of $g^m$ in terms of the order of $g$, and 4.45 characterizes when the subgroups $\langle g^m \rangle$ and $\langle g^k \rangle$ are equal. I would however recommend memorizing Corollary 4.47.

• For 4.49, you may want to use 4.45 and 4.47, but to do that, you need to select a generator $g$ for $\mathbb{Z}_m$ and translate the notation $g^k$ to additive notation. Note that $1$ is always a generator for $\mathbb{Z}_m$, and additively, $1^k$ should be written $k\cdot 1 = k$. Also, if you want another reference for modular arithmetic, you could try my Math 102 notes about congruences.

Extra practice: 4.48, 4.50, 4.51, 4.54, 4.58, 4.64(a,b,c)