It is Possible to define an isomorphism f from the multiplicative group of fourth roots of unity {1,-1,i,-i} to Z, under addition mod 4, with f(1)=1 اختر احدى الدجابات True False

Q: Suppose that G is a finite abelian group.  Prove that G has order pn where p is prime, if and only…

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Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: Prove that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.

A: If G is the simple group of order 60 That is | G | =60. |G| = 22 (3)(5). By using theorem, For every…

Q: 3*. Let (F,+) be the group of polynomials over3 X {x, a2, a3,...} be the set of all positive integer…

A: To describe the subgroup generated by the positive powers of x in the additive group F[x]

Q: No. of isomorphic subgroup of group of integers under addition is: -

A: As we know group of integers under addition is (Z,+)

Q: What is the smallest positive integer n such that there are three nonisomorphicAbelian groups of…

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Q: Let m be a positive integer. If m is not a prime, prove that the set {1,2,..., m – 1} is not a group…

A: We show that it doesn't satisfy clousre property.

Q: Prove that a group of order 12 must have an element of order 2.

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

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Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: Here is the question I'm needing help with.  Prove C*, the group of nonzero complex numbers under…

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Q: Let F denote the set of first 8 fibanacci number .Then convert F into a non abelian group by showing…

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Q: Find an isomorphism from the group G = to the multiplicative group {1, i, – 1, – i} in Example 3 of…

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

A: Solution:-

Q: prove that Every group oforder 4

A: Give statement is Every group of order 4 is cyclic.

Q: Show that the units in Commutative ring unit element forms with an abelian group wv.t multipli…

A: let the two units of ring  R are u  and v let u-1 and v-1 also belong to R. therefore, uu-1=1 and…

Q: Prove that every subgroup of the quaternion group Q8 is normal. Deter- mine all the quotient groups.

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

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

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Q: Prove that there is no simple group of order 525 = 3 . 52 . 7.

A: The prime factors of 525 are 3, 5 and 7. So there are proper normal subgroups of order either 3,5 or…

Q: 2. Use one of the Subgroups Tests from Chapter 3 to prove that when G is an Abelian group and when n…

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Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: By applying Fundamental Theorem of group homomorphism, show that the quotient groups GL(n, R)/SL(n,…

A: It is given that,  GL(n, ℝ) is the set of general linear group of all n×n matrices and  SL(n, ℝ) is…

Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

Q: 5. How many automorphisms does Klein's 4-group have?

A: No of automorphism of Klein's 4-group:  K4={e,a,b,c} f1=eabceabc = I f2=eabceacb = bc f3=eabcecba =…

Q: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

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: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

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

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Q: Suppose that o is a homomorphism from a finite group G onto G and that G has an element of order 8.…

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Q: suppose G is Finite group and FiGgH homogeneity : prove that Ir(6)|iG| Be a

A: Given that G is a finite group and F:G→H is a homogeneity, i.e. F is a homomorphism. To prove  FG |…

Q: 1. Prove Theorem 1 by setting up an isomorphism between (n + 1)-tuples and poly- nomials of degree…

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Q: Find the number of isomorphism classes of the abelian groups with order 16.

A: The order of abelian groups = 16

Q: Find a homomorphism from the group (Q, +) to (R, +). Find a one-one map if exist from Q to R that is…

A: Given :  (Q, +)  and (R, +) be the given additive groups. To find :  A homomorphism from (Q, +) to…

Q: Prove that

A: To prove: Every non-trivial subgroup of a cyclic group has finite index.

Q: ) Describe, up to isomorphism, all abelian groups of order 300 = 22 . 3 - 52 using the Fundamental…

A: Solution: Given : Group of order 300

Q: 14. Prove that the set of all rational number of the form 3"6" | m,nEZ} js a group under…

A: Denote with

Q: Prove C*, the group of nonzero complex numbers under multipliation, has a cyclic subgroup of order n…

A: n th root of unity is the cyclic subgroup of order n.

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: What is the smallest positive integer n such that there are two nonisomorphicgroups of order n? Name…

A: Non-isomorphic groups: Groups that have different Sylow-2 groups are non-isomorphic groups.

Q: Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a…

A: Consider the x is the generator of cyclic group H for xn∈H, ∅(x)=y As a result, For all members of…

Q: What is a mapping of phi defined by the automorphism of Z6?

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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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Q: Verify the corollary to the Fundamental Theorem of FiniteAbelian Groups in the case that the group…

A: To verify corollary to the Fundamental Theorem Of Finite Abelian Groups Where, G is a group of order…

Q: What is the smallest positive integer n such that there are exactlyfour nonisomorphic Abelian groups…

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Q: suppose H is cyclic group. The order of H is prime. Prove that the group of automorphism of H is…

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

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Q: Prove that there is no simple group of order 315 = 32 . 5 . 7.

A: Prove that there is no simple group of order 315=32·5·7.

Q: Suppose that H is a subgroup of Sn of odd order. Prove that H is asubgroup of An.

A: Given: H is a subgroup of Sn of odd order, To prove: H is a subgroup of An,

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