In Z, let A = <2> and B = <8>. Show that the group A/B is isomorphic to the group Z4 but that the ring A/B is not ring-isomorphic to the ring Z4.

Q: Prove that any group with three elements must be isomorphic to Z3.

A: Let (G,*)={e,a,b}, be any three element group ,where e is identity.  Therefore we must have…

Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: Prove that if x is a group element with infinite order, then x^m is not equal to x^n when m is not…

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Q: Prove that there is no simple group of order p2q, where p and q areodd primes and q &gt; p.

A: Let G be a group. |G| = p2q, where p and q are odd primes and q &gt; p. The first Sylow Theorem: A…

Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…

Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: Prove that for a fixed value of n, the set Un of all nth roots of 1 forms a group with respect to…

A: Fix an n∈ℕ. Let U=u∈ℂ | un=1 .  Note that U⊂ℂ* and ℂ* is a group under multiplication. Let u,v∈U…

Q: Suppose that f (x) is a fifth-degree polynomial that is irreducible overZ2. Prove that x is a…

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Q: Prove A3 is a cyclic group

A: We know that If G be a group of prime order then G is cyclic group

Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

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Q: If a is a group element, prove that every element in cl(a) has thesame order as a.

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Q: (d) A commutative ring R is a prime ideal of itself. (e) If p and q are primes, then there is a…

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

A: The statement is wrong.

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

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Q: 3. Let be a ring with unity and let A* be the set of all units in A. Show that the set is a group.

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Q: Let be a ring with unity and let R* be the set of all units in R.show that the set is a group

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Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

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Q: Let R be a commutative ring with unity and let U(R) denote the setof units of R. Prove that U(R) is…

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Q: - Let R be a commutative ring with unity of prime characteristic p. Show that the map op : R →R…

A: Homomorphic ring means F:R----&gt;R' and a,  b belongs to R then f(a+b) = f(a)+(b)        and f(ab)…

Q: Suppose that the fundamental group of X is Z and p(xo) is finite. Find the fundamental group of X.

A: Given fundamental group of X is  ℤ and p-1(xo) = finite value Now we have to find the fundamental…

Q: Explain why every subgroup of Zn under addition is also a subring of Zn.

A: Every subgroup of Zn under addition is also a subring of Zn as it follows the 1) Associative…

Q: Any ring R is a Jacobsen radical ring, if it is a simple ring. O مبا O

A: JACOBSON RADICAL:- The radical of the base ring R is called its Jacobson radical and denoted by…

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: 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: Show that ( Z,,+,) is a cyclic group generated by 3.

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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: Let CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

Q: The group of integers Z under addition is isomorphic to the group of rational numbers under addition…

A: Group Isomorphic: An isomorphism ϕ from a group G to a group G¯ is a one-to-one mapping (or…

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

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

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: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

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: Let S be a subring of the ring R. Thus, by definition (S, +) is an abelian group. Let 0s denote the…

A: We will take an arbitrary element from S and show that the identity property holds for both 0s…

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: 2. Suppose M and D are isomorphic groups. Prove that Aut(M) is isomorphic to Aut(D).

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

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Q: It is Possible to define an isomorphism f from the multiplicative group of fourth roots of unity…

A: since for any Isomorphism g from a group G1 to another group G2ge1=e2where e1 and e2 are indentities…

Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: Let x belong to a group. If x2e while x : x + e and x + e. What can we say about the order of x? =…

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Q: In the set of real numbers IR there is an operation defined as: I x y= Vr³+y³ %3D prove that (IR, ×)…

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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: 9. Show that the two groups (R', +) and (R' – {0}, -) are not isomorphic.

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Q: Show that the translations of Rn form a group.

A: we have to show that   the translations of Rn form a group   We know that

Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution

Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

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Q: Give an example of a subset of a ring that is a subgroup underaddition but not a subring.

A: Give an example of a subset of a ring that is a subgroup under addition but not a subring.