The group of integers Z under addition is isomorphic to the group of rational numbers under addition True False

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

Q: A group G is cyclic if and only if there exists at G such that G={a^|n  ∈ Z}. True or False then why

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Q: Let G be a finite group. Show that there exists a fixed positive integer n such that an = e for all…

A: G be a finite group.  Take any a∈G . Then {<a>} is a subgroup of G. We know by theorem order…

Q: Lagrange's theorem, in group theory, a part of mathematics, states that for any finite group G, the…

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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: Let x be an element of group G. Prove that if abs(x) = n for some positive integer n, then x-1 =…

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Q: Find a proper subgroup of the group of integers Z under addition and prove that this subgroup is…

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

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: In the group of integers Z with addition, cl(5) the conjugation class of 5 is equal to:

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Q: Prove or Disprove that the Klein 4-group Va is isomorphic to Z4.

A: The statement is wrong.

Q: Prove that Q+, the group of positive rational numbers under multiplication,is isomorphic to a proper…

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Q: Prove that any two groups of order 3 are isomorphic.

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Q: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

A: Given 'x' be an element of a group G and |x|=n. As G be a group , inverse of each element of G must…

Q: Prove that, there is no simple group of order 200.

A: Solution:-

Q: Prove that the additive group L is isomorphic to the multiplicative group of nonzero elements in

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Q: .) Prove that for every element a of a Group G, Z(G) is a subset of C(a)

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Q: Suppose that : G H is a surjective group homomorphism. Suppose that there is an element of order 27…

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

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Q: Suppose that G is an abelian group with respect to addition, with identity element 0.Define a…

A: The objective is to show that G forms a ring with respect to these operations. Since G is an abelian…

Q: Determine all elements of finite order in R*, the group of nonzeroreal numbers under multiplication

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Q: Prove or give counter example. Every finite group of order 36 has atmost 9 subgroups of order 4 and…

A: Use the third Sylow p-theorem which states that

Q: Is the set of positive integers a group under the operation of addition? Is the set of positive…

A: A set is said to be group on a binary operation if  i) It is closed ii) It is associative iii) There…

Q: Show that R* is isomorphic to G?  R* is a group under multiplication G is a group under addition…

A: To show A is one-one Let Ax1=Ax2 where x1 and x2 are two points of R*⇒x1-1=x2-1⇒x1=x2Thus the…

Q: Consider the group G={x € R such that x#0} under the binary operation *: Th identity element of G is…

A: Solution: Since for any x,y∈G, the operation * is defined as x*y=-2xy The identity element is e=-12…

Q: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: State the first isomorphism theorem for groups and use it to show that the groups/mz and Zm are…

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Q: = Prove that, there is no simple group of order 200.

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Q: Question 5. Prove that O- subgroup of itself. (the set of positive rational numbers) under…

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Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

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Q: Prove that every subgroup of Z is either the trivial group, {0}, or nZ = {nx | x E Z} for some n E…

A: To prove: That every subgroup of ℤ is either the trivial group{0} or nℤ=nxx∈ℤfor some n∈ℕ. Proof:…

Q: 9. Let G = R \ {-1} and define a binary operation on G by a * b = a +b + ab. Prove that G is a group…

A: If G is a group  it follows the following property: 1. Well defined binary operation on a nonempty…

Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: Prove that if G is a finite group, then the index of Z(G) cannot be prime.

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Q: Show that the set R of all real numbers together with usual addition of real numbers is a group.…

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Q: * Let H be a proper subgroup of the group (Z/n)*. Prove that there are infinitely many prime numbers…

A: Given H is subgroup of Z/n* Assume there exist q1, q2 with 0≤q1,q2≤1 and q1≠q2 such that q1+Z=q2+Z…

Q: Given two examples of finite abelian groups

A: Require examples of finite abelian groups.

Q: Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

A: We know that if two groups are isomorphic than they have same number of elements i.e. their…

Q: Prove or Disprove that the Klein 4-group V4 is isomorphic to Z4.

A: The Klein 4 Group is a least non cyclic group. All the none identity element of the Group, which…

Q: Consider the group G-{x eR such that x0} under the binary operation ": The identity element of G is…

A: We know that,  Every element of G must satisfy the basic condition that it should be equal to en…

Q: Let G = {2'3m5n : l, m, n E Z}. G is a group of rational numbers under the usual multiplication.…

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Q: Is the set of numbers described below a group under the given operation? Rational numbers;…

A: This is a question of abstract algebra.

Q: A ring homomorphism from a ring R to a ring S is a group homomorphism from the additive group R to…

A: We need to determine that a ring morphism from R to S is group homomorphism from the additive group…

Q: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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Q: Prove that the set of all rational numbers of the form m/n, where m, n E Z and n is square- free, is…

A: We prove it using a One-step test for subgroups.

Q: Let Q* be the multiplicative group of positive rational numbers. Prove that Q* is isomorphic to…

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Q: Let G = { a+bi ∈ ℂ : a and b are rational numbers, not both zero } . Prove that G with the operation…

A: Given - G=a+ib∈ℂ: a and b are rational numbers, not both zero.To prove-G with the operation of…

Q: 10. Prove that all finite groups of order two are isomorphic.

A: Here we use basic definitions of Group Theory .

Q: Let F be a finite field of p" elements containing the prime subfield Zp. Show that if a e F is a…

A: Given that