3330_Notes_4o1_fill

3330_Notes_4o1_fill - Math 3330 Section 4.1 Finite...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 3330 Section 4.1 Finite Permutation Groups Describing a permutation on a set with n elements { } 1234 , , , ,..., n Aa a a a a = How many permutations are there? Matrix representation of the permutation f: 123 1 1 ... ( ) ( ) ( ) . .. ( ) ( ) nn aaa a a f af a f a f a ⎛⎞ ⎜⎟ ⎝⎠ Consider the corresponding permutation on the set 1,2,…,n. 1 2 3 4 ... '(1) '(2) '(3) '(4) . .. '( ) n f fff f n In which '( ) f im = if and only if ( ) f aa = In this way, all permutation groups on sets of n elements can be viewed as the permutation group on the set 1,2,3,…,n. This group is called the symmetric group on n elements and is denoted . n S Example of a permutation on 6 elements: 123456 326541
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A CYCLE An element n f S is called a cycle if there exists a set { } 123 , , ,. .., r iii i such that 12 2334 ( ) , ( ) , ( ) ,..., ( ) r 1 fi i fi i i === = And f leaves all other elements fixed. Example The cycle notation 1 ( , , ,. .. , ) rr f iii i i = Example: The inverse of a cycle:
Background image of page 2
Not all elements of the symmetric group on n elements are cycles, but all can be written as a disjoint product of cycles.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 12

3330_Notes_4o1_fill - Math 3330 Section 4.1 Finite...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online