MATH 302 Lecture Notes - Lecture 10: Rational Number, Asteroid Family, Division Algorithm

• School: Simon Fraser University
• Department: Mathematics
• Course: MATH 302
• Professor: Jamie Mulholland

Document Summary

Def 10. 1 (group): a group is a nonempty set g, together with an operation, which can be. We will use |(cid:1833)| to denote the order of the group since this is really just the. Thm 10. 1 (uniqueness of inverses): for each element (cid:1853) in a group (cid:1833), there is a unique element (cid:1854) (cid:1833) such that (cid:1853)(cid:1854)=(cid:1854)(cid:1853)=(cid:1857) Proof: suppose (cid:1854) and (cid:1855) are both inverse of (cid:1853). Thm 10. 2 (cancellation property): in a group (cid:1833), the right- and left- cancellation properties hold: (cid:1854)(cid:1853)=(cid:1855)(cid:1853) implies (cid:1854)=(cid:1855), and (cid:1853)(cid:1854)=(cid:1853)(cid:1855) implies (cid:1854)=(cid:1855) Proof: if (cid:1854)(cid:1853)=(cid:1855)(cid:1853) then (cid:4666)(cid:1854)(cid:1853)(cid:4667)(cid:1853) (cid:2869)=(cid:4666)(cid:1855)(cid:1853)(cid:4667)(cid:1853) (cid:2869) and by associativity, (cid:1854)(cid:4666)(cid:1853)(cid:1853) (cid:2869)(cid:4667)=(cid:1855)(cid:4666)(cid:1853)(cid:1853) (cid:2869)(cid:4667). Since (cid:1853)(cid:1853) (cid:2869)=(cid:1857), then (cid:1854)(cid:1857)=(cid:1855)(cid:1857) from which it follows that (cid:1854)=(cid:1855). Def 10. 3 (order of an element): for an element (cid:1853) of a group (cid:1833) the smallest positive intefer (cid:1865) such that (cid:1853)(cid:3040)=(cid:1857) is called the order of (cid:1853).