Let G {0, 1, 2, 3, 4, 5, 6, 7} and assume that G is a group with the operation satisfying the following conditions: a a=0 for all a € G, and a b ≤a+b, for every pair a and b in G. Write down the multiplication table (Cayley table) of G.

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 3 denote the centre of a group G. then show that a 3 iff N(a) = G. also show that if G is…

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Q: Build the operation table ∗ on G = {e, a, b} knowing that (G, ∗) is a group.

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Q: Let G be a group and  a,b∈ G such that ab = ba. Let o(a)= m and o(b) = n. Show that there exists an…

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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: (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: Go Let G be a group and a, b = G. Prove that (ab)−¹ = b¯¹a-¹. Let G be a group. Prove that for all…

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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: 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 if the given set is a group defined by the given operation. Check all properties. If so, check…

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Q: Let f be the following permutation: 1 2 3 4 5 6 7 f = 4 3 6 2 1 5 7 a. Determine the group to which…

A: Given permutation is f=12345674362157 f is permutation of 7 elements so it is belong to s7. i.e. sn…

Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin…

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Q: Let G be the set consisting of the following matrices: (6 9) (6 ) ( ) -1/2 v3/2 V3/2 1/2 -1/2 V3/2)…

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Q: 4. Let G be a group and let a and b be elements of G. Prove that -1 (a*b)¹ -1 =b¹ a

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Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: x and y are two vertices of a Cayley digraph. ai’s and bj’s are generators of the Cayley digraph

Q: Determine whether or not the group (S3, ∘) is isomorphic to the group (T, ∘)

A: Two groups are given to us as ⇒S3,∘T,⊙ Therefore, both the sets S3, T are forming groups with…

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 x and g be elements of a group G. show that x and gxg-1 have the same order. now prove that xy…

A: Hence, proved.Explanation:Step 1:To prove that x and gxg-1  have the same order for any elements x…

Q: Let ℝ be the set of real numbers and let • denote the normal multiplication operation on ℝ. The…

A: Let ℝ be the set of real numbers and let • denote the normal multiplication operation on ℝ.

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: 8) Determine whether the following graphs are isomorphic. If they are, state the isomorphism f and…

A: Here, we will be showing the given two graphs are not isomorphic. We are given the graphs,

Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: Let (G, *) and (H, *) be groups. The direct product of G and H is the set G × H equipped with…

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Q: Let G be a group under multiplication, G be a group under addition and o be an isomorphism from G to…

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Q: Let G = U(32) and H = {1, 15}. The group GIH is isomorphic to one of Zg, Z, O Z, or Z, O Z, O Z,.…

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Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: It is given that x and y be two vertices of a Cayley digraph. We can write two paths as a1,a2,a3,…an…

Q: 1. Let (G, +) and (H,*) be groups. The direct product of G and H is the set G × H equipped with…

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Q: A. Let G={a,b) and examine the properties of the operation defined by the operation with aᵒa=a, a…

A: o a b a a b b b a                                                     Let   G =  a , b…

Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

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: (a) Consider (3,6) and (2,4) as elements of Z7xZ7. Compute (3,6) - (2,4). (b) Consider (3,6) and…

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Q: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

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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: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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Q: Exercise 14. Let G₁ and G₂ be a groups. Then the set G₁ X G₂ is a group with the following binary…

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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: Let G and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: let n be a fixed natural number.Verify that the set  n.Z={n.k l kEZ} is a group of (Z,+).

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