In each case, find elements a and b from a group such that |a| = |b| = 2.   a. |ab| = 3 b. |ab| = 4 c. |ab| = 5

Q: HWGrouPA Prove the fallouwing identity. secA.SecB.CSCA.CSCB SectAtB) CSCA.CSCB-SecA. secB

A: sec(A+B)

Q: Label the following statement as either true or false.   The alternating group A4 on 4 elements is…

A: We have to state whether the given statement is true or false : The given statement is : The…

Q: Which of the following groups are solvable groups: Z/nZ, D8, and S4.

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Q: Let G= How many elements does G have?  To what familiar group is G isomorphic?

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Q: 1. Consider the group D5. Give five examples of D5 acting on a set, other than the trivial action.…

A: Given: Consider the group D5. Give five examples of D5 acting on a set, other than the trivial…

Q: For any group elements a and b, prove that |ab| = |ba|.

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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: Fill in the blanks in the table below so that it becomes the Cayley table of a group with elements…

A: We will fill the Cayley table in a way such that it will form a group.

Q: 5) A. Define the dihedral group D, and write the elements. Find the order of element yx³ and compute…

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Q: Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z,…

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Q: Let X,Y be cyclic groups of order 2,3 respectively.  Find the orders of the elements of X×Y.

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: 1. Prove that in any group, an element and its inverse have the same order.

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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: 1. Write the following permutations in cycle notation. (1bd from 5.4) (1 2 3 4 5 (1 2 3 4 5 4 3 2 5/…

A: To find- Write the following permutations in cycle notation. 1234542513 1234514325

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: In the group S6, consider the elements 1 2 3 4 5 6 4 3 6 5 1 2 1 2 3 4 56 35 4 2 6 1 a = and B = (a)…

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Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

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Q: Let a be a group element such that |a| = 48. For each part, find adivisor k of 48 such thata.…

A: Given: a be a group element such that a=48.We need to find divisors k of 48 such that the subgroups…

Q: Let G={aeR: a>0}. Select all of the operations from the following list such that (G, ◊) is a group.…

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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: Let Rn be the group of rotations of a regular pentagon, and let r be counterclockwise rotation by…

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Q: Let a belong to a group and a12 = e. Express the inverse of each ofthe elements a, a6, a8, and a11…

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Q: How many subgroups of the group G = (Z30, +30)? O a. 4 O b. 7 O c. 5 O d. 6 O e. 8

A: We know that τ(n) is number of positive divisors of n Also τpaqbrc = (a+1)(b+1)(c+1) , where p , q ,…

Q: Determine which of the following are groups under the stated operations. For those which are groups,…

A: i) the pair here we draw composition table :246824826486426246886284from above table ,we concluded…

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: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

A: Group: If a set (finite or infinite) together with a binary operation satisfies the four basic…

Q: 34. Suppose a and b are group elements such that |a| = 2, b ‡ e, and aba = b². Determine [b].

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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: Let Rn be the group of rotations of a regular pentagon, and let r be counterclockwise rotation by…

A: Given that Rn be the group of rotations of a regular pentagon and r be the angle rotation by 2πn. To…