1. Show that for any permutation o, there exists a positive integer m such that om = E. Find a formula for the smallest such m in terms of the cycle structure of o.

Q: Let A = {a, b, c, d, e, f}. If the permutations P1=(b, a, e, c)  and P2=[a,b,c,d,e,f f,a,c,e,b,d]…

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Q: Show that in a group G of odd order, the equation x2 =a has aunique solution for all a in G.

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Q: Determine if g 15 is a primitive element for Z3 if the following is the output for powers of 15 from…

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Q: Let K = Q(ζ3) for a primitive cube root of unity ζ3 ∈ C Determine the splitting behaviour in K…

A: Let K=ℚζn be the given field. Where ζn is a primitive nth root of unity. Consider the prime 2. Then…

Q: ToT. (o and 8 are called conjugate elements) Let o,TE Sn. Define 6 = Show that if o (i) = j, then 6…

A: Given:

Q: If pq and r are three distinct primes and G is a group of order par, then G is not simple.

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

Q: Give a combinatorial proof of the identity (C) ) = (O).

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Q: Whether each of the following statements are true or false. Write TRUE or FALSE. (a) In the RSA…

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Q: Find all the cosets of the subgroup 4Z of Z

A: In the group of integers under addition, the subgroup consists of all multiples of 4, i.e., .A coset…

Q: Determine the orders of the elements of D∞

A: It is known that, D∞=a,b|a2=b2=e also it can be observed that, D∞=abka,abk,bak,bakb Let x∈D∞. If…

Q: Let a= (123)(145) in S6. Write a202 in disjoint cycle form. (16)(24) (32541) None of these O (14523)

A: Use the concept of permutation group to solve this problem.

Q: Suppose that <a>, <b> and <c> are cyclic groups of orders 6, 8, and 20,…

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Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: (a) Determine whether 2 is a primitive element in Z37. (b) Find an element of Z37 that has order 12.

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Q: How would we show that H is cyclic?

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Q: Suppose that {A;};eI and {B;};ej are families of nonempty dasses with the same index dass I. Prove…

A: Given that Aii∈I and Bjj∈J are families of non empty classes with the same index class I.

Q: Ifpq and r are three distinct primes and G is a group of order pqr, then G is not simple.

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

Q: entire elements of D8 in terms of products of disjoint cycles in S8 ?

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Q: .

A: To prove : The two cycles (a1,a2,...ak), (b1,b2,...bm) are disjoint cycles. If the pair of cycles ↵…

Q: Give the Cayley table for the group Z2 under multiplication modulo 12.

A: Since , We know that Z12 = 0,1,2,3,4,5,6,7,8,9,10,11 and…

Q: Q.No.4 Define Group. Also construct the Table for addition of the set of Residue class modulo 5 .

A: Group definition Let (G,+) be a binary operation such that, (a) For any a,b in G then a+b is in G…

Q: Exercise 2.4.3 (a) Find the orders of all elements of Zɛ under addition mod 8. (b) Do the same for…

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Q: a) In G = (Z54,) find [G:H] for H=. b) Let |G|=8 and G is not cyclic. Show that g4 = e for every g E…

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