(4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹. Show that a * b = b* a if and only if cd=d*c.

Q: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

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Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: Express the group in disparity

A: Topic:- range on number line

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

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Q: et (G, *) be a group. Show that (G, *°P), where x *ºp y = y*x (x, y = G), is a group.

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Q: Define on S = Q × Q* the operation (a, b) * (x, y) = (ay+ x, by). Recall that Q* = Q\{0}. 1. Explain…

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Q: If f is an isomorphism of a group G into a group H, prove that (f(a))n=e' if and only if an=e. Give…

A: Suppose f is an isomorphism from G to H, and let a be an element of G. Then: (f(a))^n = e' if and…

Q: Show that (2*,A) is an abelian group. Make sure that you prove associativity, that you identify the…

A: First to prove that the given group is abelian. Here in the given group G is the set of all subsets…

Q: Let (A, *) be a group. Let B be a subset of A such that 2|B| > a= b₁ |A|. Show that, for any a in A,…

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Q: For any two elements a and b of a group G, show that G  is abelian iff…

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Q: Show that PSL(2, Z7) = SL(2, Z7)/Z(SL(2, Z7)), which has order 168,is a simple group.

A: Given: PSL2, Z7=SL2, Z7/ZSL2, Z7 has order 168.To show that PSL2, Z7 is a simple group for that we…

Q: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

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Q: 7. Let (G, *) be a group and let a G. Define C(a)= {bEG: ab ba Z(G) = {bEG: ab-ba for all a € G}…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Consider the group G = {x E R such that x 0} under the binary operation x*y=-2xy O x*x*x=4x^3…

A: Multiplication of the elements of the group elements with respect to binary operation

Q: 8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set…

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Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

Q: Let G be a group with identity element e, and let a and b be elements of G such that a^3 = e and b^4…

A: A group is a mathematical structure consisting of a set of elements equipped with an operation that…

Q: Q1)) prove or disprove ( 1. Let (G, *) be a group, if x*y = y*x then (x*y)" = x" * y". 2. Each…

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Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

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Q: 5. The permutations a = (124)(35) and b = (12)(3)(4)(5) generate a group (G, *) of order 12. (a)…

A: Given permutation is a=(124)(35) and b=(12)(3)(4)(5) The given group G,* is of order 12. Let the…

Q: each vertex of gra 7. Prove that the order of any subgroup of a finite group divides the order of…

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Q: (a) (Z,*) with a × b = a/b is (b) (Q, *) with a * b = ab - a - b is (c) (Q,*) with a*b = a² +6² is…

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Q: 64. Express Ug(72)and U4(300)as an external direct product of cyclic groups of the form Zp

A: see my attachments

Q: Let G = (Z,, +6) is an Abelian group then how many self - invertible elements in G? (A) 1 (B) 2 (C)…

A: To solve this problem, we use the defination of group.

Q: 1. Let (G, +) and (H,*) be groups. The direct product of G and H is the set G × H equipped with…

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Q: set P(S)

A: According to the question, The power set P(S) of a set S is the set of all subsets so, of S,…

Q: If G is an abelian group, prove that (a * b)" = a" * b" for all integers n.

A: Given: "G" is an abelian group,  We need to prove that a×bn=an×bn for all integers n. Abelian Group…

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: {[a b], a,b,c,d are even Z }, A + B = A + B₁ * Let (G, *), where G = = Is G a group?

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Q: 11. Prove that the set of all 2 × 2 matrices with entries from R and de- terminant +1 is a group…

A: To show that the set of all 2×2 matrices with entries from R and determinant +1 is a group under…

Q: Let a = (17 3)(5 9 4 2) and B = (2 3)(7 4)(5 1 8) be elements of S10. Then, a'ß-1 is O (1 4)( 2758 9…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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: 5. Let G be a group and define z" z * *** *z for n factors c, where z E G and n E Z+. (a) Suppose…

A: Given that G is a group. Also zn is defined as follows. zn=z*z*z*...*z (a) Use mathematical…

Q: 7. Let (A,*) be a semigroup that satisfies the property VaEA, a*A=A and A★a=A Show that A is a…

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Q: Thus option (c) is correct. Let G be a group with G| = 96. Suppose H and K are subgroups of G with…

A: for abelian group -: condition -: 1 closure ,2 associative , 3 -: identity ,  4 -: inverse ,5-:…

Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

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Q: True or False, show justification. If a, b ∈ G, where G is a group, and n ∈ Z, show that…

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Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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Q: ^^)^^\Y) 13. If * is defined on Z such that a*b = a + b -1 for a, b E Z, show that {Z,*} is an…

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Q: 1. Show that the group (a, b, c|bª = a, c¬1 = b) is abelian. %3D

A: The objective is to show that the given group is abelian. Given group is: a,b,c|b4=a,c-1=b

Q: verity that each solution is a solution to the given it is a solution. 1. y" - 2y' + y = 0;…

A: Since there are two different questions, according to the guidelines, I am solving the first…

Q: Let S = {x €R | x + 3}. Define * on S by a * b = 12 - 3a - 3b + ab Prove that (S, *) is a group.

A: The set G with binary operation * is said to form a group if it satisfies the following properties.…