If a,b are elements in a group and a-2, b-3. them Ka)N(b=6

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

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Q: 7.)What is the center of a dihedral group Dn, when n is odd? {H,V} {Ro, R180} A В {Ro} {Ro, R90,…

A: We have to find

Q: Find each element in Dg toble. Prepore coylay Is it a group? Eaploin a

A: In this question, we find the elements of D3  For this set we need to prepare the Cayley table and…

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

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Q: o:Z, →Z, A GROUP HOMOMORPHISM 30 30 KER () = {0,5,10,15, 20, 25} o(13) = 6 now THAT, 0" (6) FIND ?

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Q: 9. Which of the following groups are cyclic? Justify your answer. a) Z;×Z12 b) Z1,×Z, c) Z2×Z8×Z16…

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Q: 4. The permutations (e, a, B, v) form a group. If e = (1)(2)(3)(4)(5)(6), a = (1)(2)(35)(46), B=…

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Q: If U(14) = , then U(14) is cyclic group generator by a) 5 b) 11 c) 4 d) None of the above

A: We have to solve given problem:

Q: 0:230 → 230 A GROUP HOMOMORPHI KER (o) = {0,5,10,15, 20, 25} %3D o(13)= 6 o"(6) FIND ? now THAT, O

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Q: 2) (b*a)1 = . If a, b are elements of a group G?

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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: Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity…

A: Calculation:Obtain the Cayley table for the set S = {5,15,25,35} under the multiplication modulo 40…

Q: 10. Given the group (S = {a,b,c, d},®) a a d ic a b a a d d The inverse of an element r, x E Sis D-…

A: Inverse element is an element which when operated with its inverse gives the identity

Q: If H and K are subgroups of G, IHI= 18 and IKI=30 then a possible value of IHNK is O 4. O 6 18 8

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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: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

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Q: Let a and b belong to a group. If la] = 24 and |b| = 10, what are the possibilities for (a) n (b)i?

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Q: Example: For a group S =że,a, a², b, ab, ab}, Fund Mhe Comyjingate Sabyimps af. = ge, a, a3…

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Q: For group elements a, b, and c, express (ab)3 and (ab-2 c)-2 withoutparentheses.

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Q: Find groups that contain elements a and b such that |a|=|b|= 2 and a. |ab| =3, b. |ab| =4, c. |ab|=…

A: Given : |a|=|b|= 2 To find:  |ab| = ?

Q: If a1, a2, . . . , an belong to a group, what is the inverse of a1a2 . . . an?

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Q: 0:Z, →Z, A GROUP HOMOMORPHISM KER () = {0,5,10,15, 20, 25} o(13)= 6 now THAT, o" (6) FIND ?

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Q: How many subgroups in Z/30Z 4 30 6 CO

A: Here we have group z/30z and find its number of subgroups.

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} where a = (1 2 3 4) and b =…

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Q: The set of integers Z with the binary operation "*" defined as a*b =a +b+ 1 for a, beZ, is a group.…

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

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Q: 10) Which of the following is not a group? * (Z,*), a* b = a + 2b Va, b E Z O (Z,*), a* b = a +b Va,…

A: Option (1) is correct.

Q: (b) Suppose G is a group, H, K < G, |H|= 30, |K| = 20, and |HN K| = 10. What is |HK|? %3D %3D

A: By theorem, Order of product of two subgroup of finite order       Let the two subgroups be H and K…

Q: Please write if the set that is given is a semigroup or a monoid or a group of none of all with…

A: Given that H be a set with a binary operation * that assigns to each ordered pair (A, B) of elements…

Q: In a group G, the law ab-Dac implies b-c is called..

A: Answer : 1 right cancelation law

Q: If U(14) = ,then U(14) is cyclic group generator by a) 4 b) 11 c) 3 d) None of the above

A: We have to find the generator of U(14)

Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…

A: According to the given information, let G be a group.

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

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Q: (c) Suppose that G = (a), a e, and a5 = e. Construct a Cayley table for the group (G,.). CIG [11

A: We shall answer first question only as you have asked more than one different question. For others…

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: Given the set S:= {2"5" : m, n E Z}. Does the set S together with multiplication form a group?…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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

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Q: Which one of the following is not a cyclic group? (a) (Z,+) (b) (2Z, +) (c) (S7,0) (d) (Z100, 100)

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Q: • Is the set Q(i) = (n+bi: n,be Q) with defined operation of addition A Nulgroup of a group (C. +)

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Q: TASK 2: Prove that if G is a group and a, b = G, then (1) o(a¹ba) = o(b) o(ab) = o(ba)

A: These two results follow by proving G is abelian. We present the detailed solution in the next step.…

Q: (c) The set K ={a'b' | 0 < i < |a| and 0 < j < |b| } is a group. Construct the complete group table…

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Q: Given the group (S = {a,b,c, d},8) %3D a d a b a b d - d The inverse of an element z, z e Sis a a с…

A: The solution is given as

Q: Please solve this Abstract algebra question.

A: Consider the provided question, (a) Given a = 6, b = 2 So, a + b = 6 + 2 = 8

Q: 13. Find groups that contain elements a and b such that lal = |b| = 2 and a. lab| = 3, b. lab| = 4,…

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Q: If H and K are subgroups of G, |H|= 20 and |K]=32 then a possible value of IHNKI is O 2 16

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Q: suppose G; A group and Hi 4 G;, For i.l,1 Give examples that violate the correctness of each of the…

A: NOTE: We cannot write a normal subgroup as numerator and group as denominator because factor group…

Q: Decide which of the elements 2,3, and 5 are generators of the group Z/6

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Q: If G is a group and a, b, c € G then (abc) 1 = a'b¯lc=1

A: Use Reversal law of inverse in group theory.