Let a and b belong to a group G. Find an x in G such that xabx-1 = ba.

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

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Q: Let G1 and G2 be groups, and let G = G1 × G2. Write e and e2 for the identities of G1 and G2…

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Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: (a) Let p: G → H be a group homomorphism. Show |p(x)| < |x| for all x E G.

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Q: I need help solving/ understanding attached for abstarct algebra dealing with permutations   Thanks

A: To prove the property of conjugation of a 3-cycle in the Symmetric group

Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

Q: For any group G, Z(G) ≤ [G, G]. Prove or give a counter example

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Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

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Q: that a and xax^-1 have same order for all x belongs G.

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Q: V Let G be a group. Show that for all a, b in G, (a*b) =b*a

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Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

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Q: For any group elements a and x, prove that |xax-1| = |a|.

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Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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Q: If F is a reflection in the dihedral group D, find all elements X in D, such that X? = F and all…

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Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: 10. Let (G, *) be a group, and let H≤ G. Define N(H) = {x € G: x¹ *H* x = H} [Normalizer of H in G].…

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Q: G let then. [b, a]= be an group and Ta %3D

A: Given that G is a group and also a,b,c∈G. To prove that b,a= a,b-1 Since G is a group, it satisfies…

Q: Let G be a group. Show that for all a,b,c G, (acb-1)-1 = bc-la.

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Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: Let G be a group and a E G. Define C(a) = {x € G|ax = xa, for all a E G}. Prove that C(a) < G.

A: A nonempty subset H of a group G is said to be a subgroup of G, if it satisfies the following…

Q: For a group G and a fixed element a ∈ G, define the subset C(a) to be the set of all elements in G…

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Q: . Let G be the additive group Rx R and H = {(x,x) : x E R} be a subgroup of G. Give a geometric…

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Q: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…

A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…

Q: Help: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be…

A: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be a…

Q: An element a of a group G has order n E z+ if and only if a" = e.

A: The given statement is False.

Q: 2. Let G be a group. Show that Z(G) = NEG CG(x).

A: Let G be a group. We know Z(G) denotes the center of the group G, CG(x) denotes the centralizer of x…

Q: Show that R* is isomorphic to G?  R* is a group under multiplication G is a group under addition…

A: To show A is one-one Let Ax1=Ax2 where x1 and x2 are two points of R*⇒x1-1=x2-1⇒x1=x2Thus the…

Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

Q: Consider the group G = {x E R such that x # 0} under the binary operation ху X * y = 2 The identity…

A: First option is correct.

Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

A: Given that G,·and H,* are two groups.

Q: For any group G, GIZ(G) is isomorphic to Inn(G)

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Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

Q: Let A be a group and let B be a group with identity e. Prove that (A x B)/(A x {e}) = B . Hint: Show…

A: Let A be a group and let B be a group with identity e. Let the operation in A is @ (say) and in B is…

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: any two elements x and y in a group G, there is a unique element z in G such that y = xz.

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Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

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Q: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…

A: Ans is given below

Q: Let G be a group and a e G such that o(a) = n < oo. Show that a = a' if and only if k =l mod n. %3D

A: Let G be a group and a∈G such that Oa=n&lt;∞. Show that ak=al if and only if k≡l mod n. If k=l the…

Q: Let H and K be subgroups of a group G and assume |G : H| = +0. Show that |K Kn H |G HI if and only…

A: Given:

Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, b eG is either a left or a right coset…

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Q: Let be a group and Ha normal subgroup of G. Show that if y.VEG such that xyEH then yx EH

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Q: F. Let a e G where G is a group. What shall you show to prove that a= q?

A: Solution: Given G is a group and a∈G is an element. Here a-1=q

Q: Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

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Q: Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

Q: Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

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Q: Let G be a group with identity e and a € G. (a) Define |G|, the order of G, and |al, the order of a.…

A: (a) The cardinality or the number of elements in a group is called the order of that group. If group…

Q: Let H and K be subgroups of a group G and assume |G : H| < +co. Show that |K Kn H G H\

A: Let G be a group and let H and k be two subgroup of G.Assume (G: H) is finite.

Q: Which of the following is cyclic group а. R b. Z С. Q d. C

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