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v43-f10snfinal

v43-f10snfinal - Notes F.P Greenleaf c 2000 2010...

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Notes: F.P. Greenleaf c circlecopyrt 2000 - 2010 v43-f10sn.tex, version 11/11/10 Algebra I: Chapter 5. Permutation Groups 5.1 The Structure of a Permutation. The permutation group S n is the collection of all bijective maps σ : X X of the set X = { 1 , 2 ,...,n } , with composition of maps ( ) as the group operation. We introduced permutation groups in Example 3.1.15 of Section 3, which you should review before proceeding. There we introduced basic notation for describing permutations. The most basic kind of permutation is a cycle . Recall that 5.1.1 Definition. For k> 1 , a k -cycle is a permutation σ = ( i 1 ,...,i k ) that acts on X in the following way (1) σ maps braceleftbigg i 1 i 2 ... i k i 1 ( a cyclic shift of list entries ) j j for all j not in the list { i 1 ,...,i k } One-cycles ( k ) are redundant; every one-cycle reduces to the identity map id X , so we seldom write them explicitly, though it is permissible and sometimes useful to do so. The support of a k -cycle is the set of entries supp( σ ) = { i 1 ,...,i k } , in no particular order. The support of a 1-cycle ( k ) is the one-point set { k } . Recall that the order of the entries in a cycle ( i 1 ,...,i k ) matters, but cycle notation is somewhat ambiguous: k different symbols obtained by cyclic shifts of the entries in σ all describe the same operation on X . ( i 1 ,...,i k ) = ( i 2 ,...,i k ,i 1 ) = ( i 3 ,...,i k ,i 1 ,i 2 ) = ... = ( i k ,i 1 ,...,i k 1 ) Thus (123) = (231) = (312) all specify the same operation 1 2 3 1 in X , and likewise ( i,j ) = ( j,i ) for any 2-cycle. If we mess up the cyclic order of the entries we do not get the same element in S n , for example (123) negationslash = (132) as maps because the first operation sends 1 2 while the second sends 1 3. In Section 3.1 we also showed how to evaluate products στ of cycles, and noted the following important fact. (2) If σ = ( m 1 ,...,m k ) and τ = ( n 1 ,...,n r ) are disjoint cycles, so that supp( σ ) supp( τ ) = { m 1 ,...,m k } ∩ { n 1 ,...,n r } = then these operations commute στ = τσ . If supports overlap, the cycles may or may not commute. Since any 1-cycle ( k ) is the identity operator, certain cycles with overlapping supports such as (4) and (345) do commute, so property (2) only works in one direction; on the other hand an easy calculation of the sort outlined in Example 3.1.15 shows that (23)(345) = (2345), which is not equal to (345)(23) = (2453). Our first task is to make good on a claim stated in 3.1.15: every permutation can be written uniquely as a product of disjoint commuting cycles . This is a great help in understanding how arbitrary permutations work. 5.1.2 Theorem. Every σ S n has a factorization σ = producttext r i =1 σ i into cycles whose supports are disjoint and fill X (3) X = r uniondisplay i =1 supp( σ i ) and supp( σ i ) supp( σ j ) for i negationslash = j 1
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Some factors may be trivial 1 -cycles, which must be written down to get the support condition (3) . The factors σ i are uniquely determined, and they commute.
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