Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 < i S n 1 such that p, > Pi+1. How many permutations in Sn have exactly one descent?
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 < i S n 1 such that p, > Pi+1. How many permutations in Sn have exactly one descent? Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 Pi+1. How many permutations in Sn have exactly one descent?
Problem 10.3. Consider the following permutation f in the permutation group Sz: f:1-3, 2 H+ 6, 3 - 3, 4 +5,5 2),6 2,7 H 1. Furthermore, it is known that f is odd. (1) Determine f by writing f as a product of disjoint cycles. (2) Determine of). (3) Compute f17 by writing f17 as a product of disjoint cycles. (4) Write f as a product of transpositions. Hint. The fact that f e Sy should narrow it down to...
4. List all left cosets of An in Sn. (See 3.7.11.) For a given permutation o in Sn, how can you tell from o which coset o An is? Example 3.7.11. Pick a positive integer n > 2 and consider the group S. We define An = {o ESO is an even permutation). We will use the first theorem above to verify that An is a subgroup of S First of all, the identity is defined to be an even...
For the permutation group of 4 elements (S4) - 1. What are its classes also find the order of each class 2.Write down the dimensions of all the irreducible representations
Abstract Algebra I 1. Write down the Cayley table for the group generated -1 0 0 1 by the matrices 1 and 1. 2. Write down the Cayley table for the permutation group generated by the permutations (12)(34) and (13) in S 4 3. What do you notice about the two Cayley tables? How do they compare with the Cayley table for Z/8Z? How about the Cayley table for the square? 1. Write down the Cayley table for the group...
0 1 1. Write down the Cayley table for the group generated by the matrices 1 0 -1 0 and 0 1 2. Write down the Cayley table for the permutation group generated by the permutations (12)(34) and (13) in S_4. 3. What do you notice about the two Cayley tables? How do they compare with the Cayley table for Z/8Z? How about the Cayley table for the square? 0 1 1. Write down the Cayley table for the group...
1. Write the permutation o = (6, 1)(4, 2)(1, 2, 3)(5,8)(1, 2) of Sg as a product of independent cycles. Is o an element of Ag? Find the order of o. Find the inverse of o. Justify your answers.
(a) Write down the definition of the inverse of an n × n matrix A. (b) Using elimination, find the inverse of the matrix I. where a, b, c, d are real numbers such that a 0 and ad -be 0.
Exercise 2. Given a permutation o E S. define a matrix P, E M. (F) by setting P.(1,j) = P.(.) = 806) 1 if i=0G) ifi 00) for all 1 Sij Sn. For example, ifo is the identity permutation, then P, (1) Show that det(P.) gn(a) for all o ES.. Deduce that the matrix P, is invertible, for all o ES (ii) Show that P.P. - Por for all 9,TES. Deduce that the matrix P, is orthogonal, for all o...