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 = Z9 x Z12 × Z16. (a) Find all subgroups of G of order 144. (b) Let H = ((3,3, 6)), the cyclic…

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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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Q: In the group Z24, let H =(4) and N= (6). (a) State the Second Isomorphism Theorem. (b) List the…

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

Q: 2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the…

A: 2. Let , the multiplicative group of non-zero real numbers.Let  be the group with operation .Prove…

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Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

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Q: Question 1 Consider the groups Z² = Z× Z and Z³ = ZxZxZ and the map ƒ : Z³ → Z² defined by the rule…

A: We know that Given groups G (with operation *) and H (with operation #), the direct product G × H is…

Q: An endomorphism of the group G is a homomorphism : G → G. Let End (A) be the set of all…

A: Using the definition of ring set of endomorphism from A is form a Ring.

Q: B (ii), (iii) (i), (ii) (i) (iv) (i), (iv) Suppose f: A B is a non-zero group homomorphism. Which of…

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Q: (B) Let G be a set of all real numbers excep: . Define * on G by a * b = |a|b, where |a| is the…

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Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

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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)…

A: Let us prove above results

Q: (c) Let G, H be groups (i) Ifo: G H is a homomorphism, then prove that o(2") = (o(x))" for all ne Z+

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Q: Let G and G' be groups. (a) If p: G → G' is a function, what is required for p to be a homomorphism?…

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Q: Let f. → (R, +> be defined by f(x) = 3x- 3 Prove or disprove that f is an isomorphism from the…

A: This is related to isomorphism

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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.

Q: (i.) Let F(x)= x². F is an automorphism on (R+,.) (j.) The set of non-zero real numbers with…

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Q: Let G and H be groups, and let 0 : G -» H be a {x E G 0(x) = e} is called the kernel of 0 and is…

A: From the given information:Let G and H be the groups

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Q: Let (G, ★) and (G,o) be groups, with respective identity elements e and ē. Suppose there is a…

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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: 4. Prove that if G and G' are isomorphic groups, then Aut(G) and Aut (G') are also isomorphic.

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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: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

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Q: If f:G\rightarrow{}G' is an isomorphism of groups, show   that the order of an element a∈G is equal…

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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

Q: An automorphism of a group G is an isomorphism G→ G. (1) Prove that Aut(G), the set of all the…

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

Q: iv Sketch the Caley Graph of the additive Group of direct product Z3× Z4 with respect to the…

A: Consider the conditions given in the question. Clearly from the hint a torus is involved in the…

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: Let f.g:(G,.) (G')and h(m) f(m). g(m), Vm € G, them h is group homomorphism if f is homomorphism O f…

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Q: G is a finite group and f be an automonphism of G satisfying the conditian f(x)= x → x=e for xEG.…

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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…

Q: 6. Let G be a group. An isomorphism from G to G is called an automorphism of G. The set of all…

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Q: An endomorphism of the group G is a homomorphism : G → G. Let End(A) be the set of all endomorphisms…

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Q: Suppose a and b are generators of a group G, that is, G = (a) = (b). Show that 4 : G → G given by…

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Q: Consider the group A₁. Let H = {e, (12)(34), (13)(24), (14)(23)}. (a) Show that H is a subgroup of…

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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