iv Sketch the Caley Graph of the additive Group of direct product Z3× Z4 with respect to the generating set consisting of a and b, where a = (1,0), b = (0, 1). (Hint: Draw it on a torus)

Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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Q: b). Let o:Z-Z be given by .0(n)=7n. Prove that o is a group homomorphism. Find the kernel and the…

A: As per our guidelines we are supposed to answer only one asked question.kindly repost the other…

Q: Show that the cyclic group of n objects, Cn, may be represented by r", m = 0, 1, 2,...,n– 1. Here r…

A: Given  Cn is a cyclic group of n elements To show Cn is generated by some element r of Cn. That is…

Q: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

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Q: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

Q: 2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its…

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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: Exercise 8.3. (a) If H₁ and H₂ are subgroups of groups G₁ and G₂, respectively, prove that H₁ÐH₂ ≤…

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Q: 3. Let G = Z96 Z24 Z6 O Z3. Find a direct sum of cyclic groups of prime power order that is…

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Q: (1) Z/12Z (2) (Zx 끄)/(6Zx 14Z) (3) (Z4 × Z4)/((2, 3)) (4) (Z4 x Z10)/((2, 4))

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Q: 3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b),…

A: 3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Q: Show that any group of order 3 is abelian.

A: The solution is given as

Q: Applying what we discussed in cyclic groups, draw the subgroup lattice diagram for Z36 and U(12).

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Q: 6. Let G be a group with the following property: "If a, b, and c belong to G and ab=ca, then b = c."…

A: For a group to be abelian it should follow that ab=ba for all a,b belonging to G. Now we will…

Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

Q: Consider the map o : G¡ → G2 defined by: 9(a) = a-! %3D (a) Does o define group homomorphism? (b)…

A: Property of homomorphism of groups....

Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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Q: 2b G = {a +b/Z:a,beQ} a additive group. b a Show that you isomorphic ? are

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Q: 4. (a) Show that every group of order 4 is isomorphic to either Z4 or V4. (b) Show that H {1,…

A: We have show given property:

Q: V2n 5) Let G be a group such that |G| = (e" xd,)!, and |H|= (n– 1), where H a %3D Subgroup of G,…

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Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

Q: 3. You have already proved that GL(2, R) = {[ª la, b, c, d e R and ad – bc ± 0} forms a group under…

A: Note: There are two questions and I will answer the first question. So, please send the other…

Q: For any elements a and b from a group and any integer n, prove that(a-1ba)n = a-1bna.

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Q: (a) What does it mean for two groups to be isomorphic?

A: see my solution below

Q: c) Show that Z,,+, is a cyclic group generated by 3

A: 3(c) To check if 3 is generator of (Z5 , +5) , we must check that 3 generates all the members of Z5…

Q: Q7/ Find all possible non-isomorphic groups of order 77.

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Q: True or False: (a) 3Z = 9Z (b)Let p be a prime number. Then Zp × Zp = Z,² (c) Every subgroup of a…

A: We have to state whether the following statements are true or false. We have  and . We have to show…

Q: 1. Show that G is closed under x. 2. Show that (G. x) in a cyclic grouP generated by t.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Use the Cayley table of the dihedral group D3 to determine the left AND right cosets of H={R0,F}.…

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Q: a 2b G= {a+bv2:a,beQ} G = additive group. a Show that you isomorphic ? are

A: Define an isomorphism between two groups

Q: Find normalizer and centralizer of the subset H = {1, r2 , r2 s} of the group D5= The dihedral group…

A: It is given that,  H = 1, r2, r2s be a subgroup of the dihedral group D5 We have to find the…

Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…

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Q: 2. Consider the groups (R, +) and (R x R, +). Define the map : RX (RX R) → Rx R defined by r(x, y) =…

A: Given: The group is, (ℝ, +) and (ℝ×ℝ, +) The map is, ℝ×(ℝ×ℝ)→ℝ×ℝ To find: a) The defined map is an…

Q: 3.10. Let G be a group containing at most three elements. Show that G is abelian.

A: Here given that G is a group contains at most three that is G contains 1 or 2 or 3 elements. Case -…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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Q: (a) Draw the lattice of subgroups of Z/6Z. (b) Repeat the above for the group S3.

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Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…

A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…

Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

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Q: Q3: (A) Prove that 1. There is no simple group of order 200. 2. Every group of index 2 is normal.

A: Sol1:- Let G be a group of order 200  i.e O(G) = 200                = 5² × 8. G contains k Sylows…

Q: (i). There is a simple group of order 2021.

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Q: o= (1,8,5,7)(2,4) y= (1,3,2,5,8,4,7,6) a. Compute oy. b. Determine whether oy is odd or even. 2.2…

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Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

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Q: The group U(15) is an internal direct product of the cyclic subgroups generated by 7 by 11, U(15) =…

A: We  have to check that U(15)= <7>×<11>  Or not. Concept: If  n =p1.p2 Where p1 and p2…

Q: 3. Let Dg be the group of symmetries of a regular octagon. No element of Dg of order 2 is an element…

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Q: a 26 G= {a+bV2:a,beQ} G = additive group. a Show that you are isomorphic ?

A: Two given additive groups are G=a+b2:a,b∈ℚG'=a2bba:a,b∈ℚ Let us consider a map φ:G→G' defined by…

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: Find the order of each element of the group Z/12Z under addition

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Q: (a) Give the definition of a gyclic group. (b) Prove that every eyclic group is abelian . (c) Prove…

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Q: (a) In the group , find ([2]) and then find the order of the quotient group Z₁0/([2]). (b) Prove or…

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