a) In G = (Z54,) find [G:H] for H=< 24 >. b) Let |G|=8 and G is not cyclic. Show that g4 = e for every g E G.

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Q: Exercise 8.2. Let a e Sym (10) be given by α= = (1357) (126) (379) (0543) Write a as a product of…

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Q: 12. Describe the group of the polynomial (x* – 5x? + 6) € Q[x] o over Q.

A: please see the next step for solution

Q: Exercise 8.4. Suppose o e Sym(n) is a k-cycle. Show that TOT-¹ is also a k-cycle for every 7 €…

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Q: Theorem 2.1 For every graph G, o (G) ≤ x (G). Proof It suffices to assume that G is a nontrivial…

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Q: in fuzzy graph theory what is the CONN And give examples

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Q: Prove that the chromatic number of G is 4. Ie: x(G) = 4.

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Q: 5*. Suppose that 0: G -+ H is a homomorphism. Prove that (a) (a")= (x)" for all positive integers n.…

A: To prove the properties of group homomorphisms

Q: B A F (Ε

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Q: 4 Let f = (4 5 6) and g = (1984) (2 7 5) (36) be two permutations in S9. (a) Compute fogo f-1, and…

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Q: Prove: Let F,F'be forests on the same set of vertices, with |E(F)|< |E(F')|. Show that F'has an edge…

A: Given below clear explanation

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Q: Prove part (A) by filling in the blanks. Let σ ∈ SX be an arbitrary cycle of length k. Then σ can be…

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Q: 5.26 Prove that if G is a graph of order n such that F(G) > (n – 1)/2, then 2(G) = F(G).

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Q: Prove that the order of a k-cycle in S, is k.

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A: denotes symmetric group.

Q: 4. If G = (Z8,) then G has an element of order 5. *True *False

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Q: How would we show that H is cyclic?

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Q: Please do the folloowing questions with handwritten working out

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