Exercise 4. Consider the permutation group S7. a. Show that the subgroup generated by the element...
Let wE S7 be a permutation which rearranges 7 objects as follows, showing the result on the lower line 2 3 4 6 7 5 5 4 2 7 6 1 3 a) Express was a product of disjoint cycles representing how each object moves Is w an even permutation, or an odd permutation? What is its order? products of disjoint cycles b) Calculate w3, w5 and w' 2 as c) Does there exist T E S7 for which T-lwr...
2. Consider the permutations a (123)(45) and b (2543) in the symmetric group S (a) Compute the conjugate permutation ca using (i) the definition a-b ab (b) What is the order of a? How many permutations have the same shape as a; that is, (x x x)(x x). (c) What is the subgroup H of all permutations in Ss that commute with the permutation a? d) Using the result of the previous part, or otherwise, find 5 other permutations bi,...
This is all about abstract algebra of permutation group. 3. Consider the following permutations in S 6 5 3 489721)' 18 73 2 6 4 59 (a) Express σ and τ as a product of disjoint cycles. (b) Compute the order of σ and of τ (explaining your calculation). (c) Compute Tơ and στ. (d) Compute sign(a) and sign(T) (explaining your calculation) e) Consider the set Prove that S is a subgroup of the alternating group Ag (f) Prove that...
Please answer all the four subquestions. Thank you! 2. In this problem, we will prove the following result: fG is a group of order 35, then G is isomorphic to Z3 We will proceed by contrd cuon, so throughout the ollowing questions assume hat s grou o or ㎢ 3 hat s not cyc ić. M os hese uuestions can bc le nuc endent 1. Show that every element of G except the identity has order 5 or 7. Let...
Which of the following statements are true? The element (1,2,3,4,5) € Shas order 5. The element (1, 2, 3)(4,5) € S has order 6. The element (1,2,3) (4,5,6) E S6 has order 6. The element (1, 2, 3)(4,5)(6, 7, 8, 9, 10) E S5 has order 30. Which of the following statements are true? The elements (1,2,3)(4,5) and (1, 2)(3, 4)5,6) are conjugate in Se. The element (1, 2, 3, 4, 5, 6, 7)(8,9,10)(11, 12) is an even permutation. The...
Please answer the parts 6 and 7. Thank you. 2. In this problem, we will prove the following result: If G is a group of order 35, then G is isomorphic to Zg We will proceed by contradiction, so throughout the following questions, assume that G is a group of order 35 that is not cyclic. Most of these questions can he solved independently I. Show that every element of G except the identity has order 5 or 7. Let...
5) Leth-{ơes,lo(4) 4) That is, H is the set of permutation in S4 that leave the element 4 in its place. (i) Prove that H is a subgroup of S4. (ii) Prove that S is isomorphic to H. Explicitly give an isomorphism f: S3 → H listing the 6 elements of S, and giving the permutation in H to which it is sent under f. (ii) 1S "Spot check" the homomorphism property by showing that 5) Leth-{ơes,lo(4) 4) That is,...
a = 14. Consider the permutations (13)(24)(56) and B = (1 4)(26)(35) expressed in cycle notation. How many permutations y of {1,2,3,4,5,6} exist with the property B = y lay where we compose from left to right? (b) 48 (c) 6 (d) 24 (e) 3 (a) 8 1 15. You are given that {a,b,c} = {1,2,3} and, from the 8-puzzle 2 3 4 5 6 7 8 the following configuration has been reached by moving squares in and out of...
The following questions pertain to permutations in S8 (a) Decompose the permutation (1 2 3 4 5 6 7 %) into a product of disjoint 13 6 4 1 8 2 5 7 cycles. = (b) Decompose the permutation T= (1,4, 3) (5,7,6,8) into a product of transpositions. (c) Determine whether o and T are even or odd permutations. (d) Compute the product OT.
Problem 4. Let G be a group. Recall that the order of an element g G is the smallest k such that gk = 1 (or 00, if such a k doesn't exist). (a) Find the order of each element of the symmetric group S (b) Let σ-(135)(24) and τ-(15)(23)(4) be permutations in S5. Find the cycle decompositions for (c) Let σ-(123456789). Compute ơ-i, σ3, σ-50, and σί006 (d) Find all numbers n such that Ss contains an element of...