Read: Finish Section 2.2 and start Section 2.3.
Turn in: 2.31, 2.32, 2.35(b,c), 2.36(a,b,c)
- Although I’m not asking you to prove 2.29, please read it over, and know that you can use it in the future. That theorem is something proved in our Math 108 course.
- About the definition of a group: Axiom 0 refers to “closure” which was defined in the discussion following Definition 2.15. It is only for emphasis since if you already know $*$ is a binary operation on $G$ then Axiom 0 must hold.
- About 2.35: please make sure to explain why each of the group axioms is true. Important: remember to use previous work. For example, in each case of 2.35 we already discussed in previous problems that composition is a binary operation on the set. Thus, it only remains to check Axioms 1,2,3. Also, since the operation is function composition, you can use 2.29 for associativity. So really the work is to identify an identity element (satisfying Axiom 2) and to identify an inverse $g’$ of each element $g$ (satisfying Axiom 3).
- About notation: in this book, $\mathbb{Z}$ refers to the integers and $\mathbb{N}$ refers to the positive integers (i.e. $\mathbb{N} = \{1,2,3,\ldots\}$).