Exercise 4.1. (1) Let : G → G' be an isomorphism between two groups G and G'. Suppose G is cyclic. Show that G' is also cyclic. (2) An automorphism of a group G is an isomorphism from G onto itself. Determine the number of automorphisms of the following groups (i) (Z,+) (ii) (Z15, +15).

Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

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Q: 2. Let G and H be groups, and let a: G H be a homomorphism. Prove the following statements. (a) a(G)…

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Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

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Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: Let (G, *), (G', *' and (G", ") be groups, and let :G G' and : G' → G" be isomorphisms. Prove that t…

A: We will use the definition of isomorphism to prove that the composition of two isomorphisms is again…

Q: (6) Let G be a noncommutative group. Show that G has a nontrivial subgroup.

A: We recal Cauchy's theorem and then use it

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A: As per the policy, we are solving 3 subparts of the given question. Please repost the question and…

Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: 1. (10 points) Show that if ƒ : G → H is an isomorphism of groups G and H then ƒ-l (the inverse…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

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Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | |   2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…

Q: 2* Let f G H be a group homomorphism. Prove that if x E G and n is a natural number then f(x)= f(x)"

A: To prove the required property of group homomorphisms