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: 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: 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: 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: How would we show that H is cyclic?

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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: 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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