Consider the group U(12) = {1, 5, 7, 11} with multiplication modulo 12. (a.) Is U(12) isomorphic to ZĄ? Justify your answer. (b.) The function : U(12) U(3) kk mod 3 is a group homomorphism. (You do not need to check this.) What is ker(p)? (c.) Is H a normal subgroup of G? Justify your answer.

Q: (ii) Let K={(), (1 2), (3 4), (5 6), (1 2)(34), (1 2) (5 6), (3 4) (5 6), (1 2)(3 4)(5 6)} with the…

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Q: (d) Show that Z under addition is a subgroup of under addition

A: Given : ℤ is a set of integer. To prove : ℤ is subgroup of Q under addition.

Q: Which of the following is correct? * O U(14) is isomorphic to S3 O U(14) is isomorphic to U(12)…

A: (a) U(14) is isomorphic to S3. Consider U(14). The elements in U(14) are the elements that are…

Q: (b) Is Z7 Z10 a cyclic group? Explain. (c) Find the order of the element (2, 2) in Z7 Z10.

A: Cyclic groups and order of an element

Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

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Q: Let p: Z → Z be a group homomorphism given by the rule o (x) = 10 . x %3D for any a E Z. a. Compute…

A: Given : φx=10.x Therefore, φ-8=10.-8=-80. Now, x can be positive as well as negative. Therefore,…

Q: 1. Let A be an abelian group, and let G = A × A (Cartesian product), which also is a group. Let 6:…

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Q: 2. Let G be the group of 2 x 2 matrices under addition and H = {(ab): a +d=0}. Prove that H is a…

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Q: Let n > 2 be an integer, and let X C Sn × S, be the set X = {(ơ, T) | 0(1) = T(1)}. Show that X is…

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Q: Q21 Let G={xER:x6=-1}. Define A on G by xAy=x+y+xy. (a) Show that gives a binary operation onG. (b)…

A: Introduction: Group is an algebraic structure in modern algebra. When a set along with a binary…

Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

Q: (True/False) Suppose G is a group of order 25. Then every proper subgroup of G is abelian.…

A: (a) We know any group of order p2 is abelian where p is a prime number. Also, any subgroup of an…

Q: Suppose that the fundamental group of X is Z and p(xo) is finite. Find the fundamental group of X.

A: Given fundamental group of X is  ℤ and p-1(xo) = finite value Now we have to find the fundamental…

Q: (b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N. Let m and n be…

A: We make use of the gcd to answer this question. DEtails are given in step 2.

Q: Updated the introduction of the problem. It was messed up. Introduction of the problem: Let G = D6…

A: To find the order of LR60 in G, we need to compute (LR60)^n for different values of n until we get…

Q: How would you solve such a question: Suppose f : Z → Z5 is a group homomorphism, and suppose f (3) =…

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Q: Let P₂(x) = {ax² + b | a, b e R and a #0}. If p(x) = ax² +b and g(x) = cx² + d are elements of…

A: In the given problem we have to show the the set p2(x) is a group with respect to the binary…

Q: Are the groups (co ટીડ, defined in class, as answer.

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Q: Let G be the group of all transformations on R which have the form x → ax + b where a, b E R, a ± 0.…

A: Let us assume f(x)=ax+b a,b ∈ℝ  and G is the collection of all such f.

Q: Which of the following sets forms a subgroup in the additive group Z? (a) {1} (b) N (c) {1, -1}. (d)…

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Q: 1*. Let f G H be a group homomorphism. Prove: (a) If l is the identity of G then /(lg) is the…

A: To prove the required properties of a group homomorphism

Q: Question 10. Prove in detail that G(T) is a subgroup of G.

A: Here we use one step subgroup test

Q: Find the orders of each of the elements of U(14). Show that it is cyclic and find all its…

A: Given :-  The orders of each of the elements of U14. To Determine :- It is cyclic and all its…

Q: 1. Let (G, *) be a finite group of order n. Then (a). Let n = pq (p, q are a prime numbers) and let…

A: This is a problem of Group Theory.

Q: Let G be a group and let a e G. In the special case when A= {a},we write Cda) instead of CG({a}) for…

A: Consider the provided question, According to you we have to solve only question (3). (3)

Q: Let K={β∈S5∣β(1)=2 and β(4)=4}K={β∈S5∣β(1)=2 and β(4)=4}. Prove that KK is not a subgroup of S5.

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Q: 2. Let (G, *) and (G', $) be groups, and let : G→ G' be a homomorphism. Let K be a subgroup of G'.…

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Q: 42. Let G be the group of 2 × 2 matrices under addition and b •{(a $): a +d=0}. d H = Prove that H…

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Q: 1. Let A be an abelian group, and let G = A x A (Cartesian product), which also is a group. Let 6:…

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Q: If H and K are subgroups of G, IH|= 16 and |KI=28 thena possible value of |HNK| is 8. 6. 16

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Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: (True/False) Suppose G is a group. Then : G x G →G, (x,y) → ay is a group homomorphism. (True/False)…

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Q: If G and H are groups with identities 16 and ¹H, respectively, and f:G ---> H is a homomorphism,…

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Q: Show that the group (U7, *) is isomorphic to {Z6, +}. how that there is an isomorphism between the…

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Q: Consider the two multiplicative groups (R.) and (R;. .) and let f: (R.) → (R.) be a map defined by…

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Q: Problem 3. Prove that if m ± 1n, then the symmetric groups S and S. are not isomorphic.

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Q: 4. Given G and G' be two groups with respect to multiplication operation and 0: G→ G' is a…

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Q: 14. Prove or disprove (a). (H.)( (G, *), and H is an abelian subgroup ⇒ HAG. (b). (H,*) ≤ (G,*), and…

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Q: (4) Let G be a group. Prove that Z(G) is a normal subgroup of G.

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Q: 5. Decide if each of the following sets H is a subgroup or not of the group G =R x R, where the…

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Q: Let N be a G-space, then (i) if a, ß E N and ß = a*, then Gg = x-Ggx. (ii) if 0:N → N' is a…

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