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Permutations. November5,2005The purposeofthis note istosummarizesomepropertiesofthepermutations.

Recallthatapermutationofnisastring=(1, . . . , n),wherei {1, 2, . . . , n}=j isiandi =j. Abetterdefinitionistosaythatapermutationisabijective

function:{1, 2, . . . , n} {1, 2, . . . , n}, (i)=i.

DenotethesetofallpermutationsofnbySn. AdistinguishedelementofSn istheidentitypermutation,=(1, 2, . . . , n).

If=(1, . . . , n) Sn,wesaythat(i, j)isaninversionifi >j andi

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torows instead ofcolumns. But that definition will not behavewellwith respecttomultiplication.

Verify that sg() =det(P)andP =P P.Fromthis, it also followsthatsg()=sg()sg()andthatsg(1)=sg(). Inparticular,theproductoftwoevenpermutationsisanevenpermutation,andtheinverseofanevenpermutationis alsoeven. So the evenpermutations form a subgroup of Sn (note that the oddpermutationsdont).

5. Animportantclassofpermutationsisthetranspositions. Apermutation iscalledsaidtobeatransposition(ij), ifi =j,j =iandk =k, forallk=i,j.Notethateachtransposition is itsowninverse. Their importancecomesfromthefact that every permutation can be written as a product of transpositions. Thisfactisnothardtoproveby induction. The ideaisthat,givenapermutation ofn,the numbern appears in some position in , sayj =n. Thenbymultiplying by transpositions we can move n to the nth position and regard the resultingpermutationas apermutation of n1. Then apply the induction. It wouldbe agoodexercisetotrytowritedownthisideaintoaformalproof.

Forexample=(4,2,3,1)=(14)and =(3,1,2,4)=(23)(12).There are manyother beautiful and important propertiesof permutations (for

example, the cycle decomposition of a permutation), but we will leave them forsomeothertime.