(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: 4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and…

A: and are group. be a group isomorphism.We have to prove that if  has finite order the  has finite…

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

A:

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

A:

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…

A:

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

A:

Q: (4) Let G be a group, a, b E G such that |a| = 20, |b| = 12. Suppose (a) (b) ‡ {e}. Prove that b²a¹0…

A: <a>∩<b>≠e<a>∩<b> order either 2 or 4if (<a>∩<b>) has order 2then…

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…

A:

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: 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 (G, *) be a group. Prove the following. (a) For each y = G, (y-¹)-¹ = y. (b) For x, y = G,…

A:

Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

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: Let x, y be elements in a group G. Prove that x^(−1). y^n. x = (x^(−1).yx)^n for all n ∈ Z.

A: