Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

Q: Let A = {a, b, c, d, e, f}. If the permutations P1=(b, a, e, c)  and P2=[a,b,c,d,e,f f,a,c,e,b,d]…

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Q: Which image does not belong to the group? a. b. C. d.

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Q: Which of the following groups are cyclic?a. U(35)b. U5(40)c. U8(40)

A: Given: (a) U35(b) U540(c) U840 We have to check which one of the given groups is cyclic.

Q: This means we start with a set of elements and a rule for combining the elements. From there we say…

A: The group is closed under its operation. Consider the set G and two elements a, b in it. If the set…

Q: 4.12 write the group table for 2. U(40)/(3)

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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 a be an element of order 8 in a group G. 6 Then order of a Select one: O a. 6 о Б.48 C. 2 d. 4

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Q: (c) Let G be a group and a, b € G be elements of G each of order 3 such that a² = 6² and prove tha a…

A: Given: A group G, and a,b∈G order of a and b is 3 and a2=b2. To Prove: a=b.

Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…

Q: 4.1. Let a and b be elements of a group G. Assume that a has order 7 and that a³b = ba³. Prove that…

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Q: mat do you call a group that is commutative?

A: A detailed solution is given below.

Q: In each case, find elements a and b from a group such that |a| = |b| = 2.   a. |ab| = 3 b. |ab| =…

A: a=2b=2

Q: Check whether the following are groups. Justify your answer. (i) (G, *) where G= {1, 2, 4, 8}, and *…

A: To check G is a group under the given operations * and ∇. G,· is a group only if it satisfies the…

Q: Consider the group G=A4. List the elements of A4 in cycle notation, and determin the order of A4.

A: Consider the group G=A4.  We have to list the elements of A4 in cycle notation, and we have to…

Q: Suppose the table below is a group table. Fill in the blank entries. a b c d e e e a e b d e d a d |…

A: The given table,   e a b c d e e         a   b       b   c d e   c   d   a b d…

Q: How many elements of order 5 might be contained in a group of order 20?

A: using  third Sylow Theorem

Q: The inverse of each element in the group ({1, 3, 5, 7}, ∙8)

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Q: 6. Complete the multiplication table for the group G = {a, b, c, d} a d a b d

A: QuestionDetails: Part of the multiplication table forthe group G = {a,b,c,d} is given. Complete the…

Q: The following is a Cayley table for a group G. The order of 3 is: 2 4 2 4 2 3 4 1 3 4 5 2 4 ere to…

A: Given Cayley table

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

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A: Given set is G = <a, b | a^6 = b^3 = e, b^-1ab = a^3>.

Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

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Q: Show, by example, that in a factor group G/H it can happen thataH = bH but |a| ≠ |b|.

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Q: (1).* 4 and |6| = 2, Let G be a group. Let a, b EG such that |a| and a b = ba. Then determine |ba|,…

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Q: 4.3. Let a and b be elements of a group G. Prove that ab and ba have the same order.

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Q: 10. Let a be an element of a group. a) Suppose a = 12. Show that a = 12 or a = 60. %3D %3D Suppose a…

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Q: Write down the six elements of S3 and give the multiplication table for the group. The six elements…

A: Given six elements of S3 are     1=123123, a=123231=123 , b=123312=132, c=123213=12, d=123321=13,…

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

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Q: 2.6. Let G be a group. Define an opposite group Gº with law of composition a * b as follows: The…

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Q: 6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10x Zgs c) Z4xZ2sx Z, d) Z22XZ21 xZ6s

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Q: Prove that the inverse of any element in a group is unique

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Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________