Homework 16

Read: Continue with Section 4.1.

Turn in: 4.18, 4.20 4.24, 4.25, 4.27

• You should be able to make use of 4.17 to prove 4.18

• Note that 4.24 is an “if and only if” statement, so there are two things to show. For the “backward” direction, you will assume that $n$ divides $i-j$; what does this imply about $g^{i-j}$? For the “forward” direction, you assume $g^i = g^j$ and want to prove that $n$ divides $i-j$. One strategy for proving that $n$ divides $i-j$ is to use the division algorithm to write $i-j = qn +r$ for some $0\le r < n $ (note that $r$ represents the remainder when dividing $i-j$ by $n$); then prove that $r = 0$.

• For 4.25, you should use 4.24 (even if you don’t complete the proof of 4.24).

Extra practice: 4.19, 4.21, 4.22, 4.23, 4.26

• Challenge problem (just for fun): In 4.20, you proved that if $G$ is finite, then $\forall g\in G$, $\exists n\in \mathbb{N}$ such that $g^n = e$. What happens if you switch the order of the quantifiers? Try to prove this: if $G$ is finite, then $\exists n\in \mathbb{N}$ such that $\forall g\in G$, $g^n = e$. Do you see the difference?