9-17 - 9-22 Feedback
• the nonexample for abelian is a nonexample of a group, but not really a good nonexample for abelian. Find a nonabelian group.
• HW3 Problem 9(a) has some typos in the proof of onto.
• the definition of cyclic should include the form of a cyclic group ($\langle a \rangle = \{a^n \ | \ n \in \mathbb{Z}\}$
• on HW4 Problem 7, you have to be careful. $a^j\in G$ for all $j \in \mathbb{Z}$, so you can’t say there’s a max $j$ for which $a^j\in G$, instead, if you look at the suggested set in the hint, you may conclude that two of the elements in the set must be equal and continue as you have in the proof. In the last line, $n=x-y$ is the integer that works for the $a\in G$ taken at the start of the proof. You might need a different integer for a different element of $G$.
• the definition generate has a typo, the example should be for $\mathbb{Z}_{4}$, not $\mathbb{Z}$
• HW4 Problem 5, for the identity and inverses, you should check that it’s a left and right identity (and inverse)
• the order of a group is in the glossary; someone should also add the definition of the order of an element.
• Hw4 Problem 1 part (a), $a\times b$ might be much bigger than $n$, so you might need to subtract more multiples of $n$ from the product
• HW3 Problem 7 1-1 uses the fact that $\psi$ is 1-1 also; onto uses the fact that the other two functions are onto. You are missing that word in the proof. Eqn 1 has a typo (missing $\psi$)
• Theorem 5.14 has some typos in the statement