Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is

A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…

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: Let H be a subgroup of a group G and a, bEG. Then be aH if and only if * a-1b e H O None of these…

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

Q: If a, b are elements in a group G, show that (ab)-1= b-1a-1

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Q: true or false, Let a and b be elements of a group G. If a = a −1 and b = b −1 , then ba is the…

A: Given a and b be elements of a group G. a = a−1 and b = b−1

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: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

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

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Q: (G,+) is a finite group such that (a+b)^2 = a^2 + b^2 for all a,b E G. show that G is abelian

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Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: For any elements a and b from a group and any integer n, prove that (a21ba)n 5 a21bna.

A: For any elements a and b form a group and any integer n, We have to prove that a-1ban=a-1bna.

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

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

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Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

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Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Let G be a group and let H< G. If [G: H] = 16 and |H| = 21, then what is |G|?

A: The expression, G:H can be written as GH .

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: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

Q: Consider the group (Z,*) defined as a*b=a+b , then identity (Neutral) element is     a 1   b -1…

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Q: . Let G be a group and a e G. An element be G is called a conjugate of a if there exists an element…

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Q: Let a and b belong to a group. If la| = 12, \b| = 22, and (a) N (b) + {e}, prove that a6 = bl1.

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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: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

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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: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove

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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 a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

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: 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: If a is an element of order 8 of a group G,

A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e  where e is an identity…

Q: Let a,b be elements of a group G .Assume that a has order 5 and  a^3b=ba^3. prove that ab=ba.

A: Let a,b be elements of a group G .Assume that a has order 5 and a^3b=ba^3. prove that ab=ba.

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: Let G be a group, and let H < G. Assume that the number of elements in H is half of the number of…

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Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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Q: et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an…

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Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')

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Q: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.

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Q: Let g be an element of a group G such that x 2 = g and x 5 = e. Then solve for x in terms of g.

A: Given : x² = g and x⁵ = e

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: 5. Let G be a group and let a € G. An element b E G is called a conjugate of a if there exists an…

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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: let G be a group, a,b E G  such that bab^-1 =a^r , for some  r E N, where N are the natural ones,…

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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: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

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

A: