Exercise 14. Let G₁ and G₂ be a groups. Then the set G₁ X G₂ is a group with the following binary operation: (a, b) * (c,d) = (ac, bd). The group G₁ X G₂ is called the direct product of G₁ and G₂.

Q: Is the set ℤ+ under addition a group? Prove your answer using the properties of the group. Note:…

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Q: 5. Let G be the multiplicative group {[2], [4], [6], [8]} = Z10-

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Q: (b) Is Z7 Z10 a cyclic group? Explain. (c) Find the order of the element (2, 2) in Z7 Z10.

A: Cyclic groups and order of an element

Q: Suppose that |a| and |b| are elements of a group and a and b commute.If |a| = 5 and |b| = 16, prove…

A: Given, a and b are elements of a group such that a and b commute.Also, given |a| = 5 and |b| = 16.…

Q: Prove that a group of order 28 must have an element of order 2.

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Q: Prove that a group of order 63 must have an element of order 3.

A: To Prove: A group of order 63 must have an element of order 3.

Q: -) Write the Cayley table for the group U(15).

A: U(15) is the set of all elements in ℤ15={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} ; x≡x ,such that…

Q: In the frieze group F7, write x-3zxyz in the form xnymz k.

A: It is known that F7=xnymzk∥n∈Z,m=0,k=0 or k=1 and x is the translation of R2, and it map u,v to…

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: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: For any two elements a and b of a group G, show that G  is abelian iff…

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

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

Q: Prove that a group of order 28 must have 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 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: 2. a) The operation in a finite group can be specified by writing down a table. Write the table for…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Let P₂(x) = {ax² + b | a, b e R and a #0}. If p(x) = ax² +b and g(x) = cx² + d are elements of…

A: In the given problem we have to show the the set p2(x) is a group with respect to the binary…

Q: In the frieze group F7, write x2yzxz in the form xnymz k.

A: It is given that, F7=xnymzk|n∈Z, m=0 or m=1, k=0 or k=1. we need to write x2yzxz  in the form of…

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: Show that Z2 is the only group that has exactly two conjugacy classes.

A: Given a group Z2.We have to show that, Z2 is the only group that has exactly two conjugacy classes.

Q: Let α = (12)(34) and β = (24). Show that the group generated by αand β is isomorphic to D4.

A: It is given that α=1234 and β=24. It is required to show that the group generated by α and β is…

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: Write Q(sqrt2) for the set described in exercise 3.1 (iii). given a non zero element a + bsqrt(2) of…

A: It refers to the set Q(2), which is the set of all numbers of the form a+b2, where a and b are…

Q: Show that Z2 is the only group that has exactly two conjugacy classes

A: Here we know that, ℤ2=0, 1. To show that ℤ2 is the only group that has exactly two conjugacy…

Q: 3. Let {: ) G = | a,be Q, a² +b² # 0 26 a Determine if G is a group with respect to matrix…

A: Satisfy all four properties for proving G be a group.

Q: Let Dn be the dihedral group of order 2n. Decide whether the following pairs of groups are…

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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: Explain why the group (Z4, +) is NOT isomorphic to the group (LI, ∘) Group (LI ∘): The symmetries…

A: If two groups are isomorphic then they should have same number of elements

Q: 2 3 4 1 4 3 2 3 4 1) 1 2 2. Let X = {1,2, 3, 4}. Consider the following elements of P(X): a = (; 2 3…

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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: 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…