MATH 302 Lecture 13: 13

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

Document Summary

Is precisely the set of numbers that would appear as 1-cycles in the disjoint cycle form of (cid:1865)(cid:1867)(cid:1874)(cid:4666)(cid:2009)(cid:4667)=(cid:1858)(cid:4666)(cid:2009)(cid:4667) Commutators are useful in mathematics wherever non-commutative operations occur. Def 13. 1: if (cid:1859), are two elemtns of a group , then we call the element the commutator of (cid:1859) and . For a permutation (cid:2009) (cid:1845) define the fixed set of (cid:2009) to be the set of all numbers in [(cid:1866)]= {(cid:883),(cid:884),(cid:885), ,(cid:1866)} that (cid:2009) doesn"t move: (cid:1858)(cid:4666)(cid:2009)(cid:4667)={(cid:1865) [(cid:1866)] | (cid:2009)(cid:4666)(cid:1865)(cid:4667)=(cid:1865)} (cid:2009) The set of numbers that are not fixed by (cid:2009), the ones that are moved: ={(cid:1865) [(cid:1866)] | (cid:2009)(cid:4666)(cid:1865)(cid:4667) (cid:1865): moved set of (cid:2009, numbers that appear in cycles of length (cid:3410)(cid:884) For a subset (cid:1827) [(cid:1866)] and a permutation (cid:2009) (cid:1845), we denote the set of all images of the elements of (cid:1827) under (cid:2009) as (cid:2009)(cid:1827): (cid:2009)(cid:1827)={(cid:2009)(cid:4666)(cid:1865)(cid:4667) | (cid:1865) (cid:1827)} For (cid:2009)=(cid:4666)(cid:883) (cid:889) (cid:885) (cid:886) (cid:883)(cid:884)(cid:4667)(cid:4666)(cid:887) (cid:891)(cid:4667) (cid:1845)(cid:2869)(cid:2871), the set of objects that are moved is (cid:1865)(cid:1867)(cid:1874)(cid:4666)(cid:2009)(cid:4667)=