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

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: Exercise 3.2.6 Show that if G and H are isomorphic groups, then G commutative implies H is…

A: A group G is called Commutative if for any a,b in G imply ab=ba

Q: 9. Prove that a group of order 3 must be cyclic.

A:

Q: Q3\ Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

Q: Theorem Let f: G H be a group homomorphism. Then, Im f≤ H.

A: Let us consider the mapping f:G→H . Then f is group homomorphism if  f(x·y)=f(x)·f(y) where, x,y∈G.…

Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

Q: 3. Prove that the groups (R.) and (R,+) are not isomorphic, where (R* R{0},-) is the multiplicative…

A: We have to show that the groups ℝ, + and ℝ*, · are not isomorphic. Here, the group ℝ*, · does not…

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: What is the automorphism group of ?A ut (Z3, +)

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

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Q: If Φ is a homomorphism from Z30 onto a group of order 5, determinethe kernel of Φ.

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Q: Exercises 3(a) Prove that the groups $ and D6 are isomophic ; (e) Preve that the additive and the…

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Q: Prove that if x is a group element then |x| = |x-1|.

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

A:

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: (a) What does it mean for two groups to be isomorphic?

A: see my solution below

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

A: Solution:-

Q: Prove or give counterexample. For any group G, Z(G) ≤ [G, G].

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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: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A:

Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

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: Let G be any group with the identity element e. With using the Group Homomorphism Fundamental…

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Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

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

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Q: Let G1 and G2 be any two groups and θ : G1 → G2 be a group isomorphism. Let H1 ≤ G1. Prove that H2…

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Q: Show that the multiplicative group Zfi is isomorphic to the additive group Z10.

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Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: Let G be a group and let Z(G) be the center of G. Then the factor group G/Z(G) is isomorphic to the…

A: We have to check G/Z(G) is isomorphic to group of all inner automorphism of G or not. Where, Z(G) is…

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 40Z {40x | * € Z} is a subgroup of the group Z of integers. Note: Z is a group under the…

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Q: 17. Let f: be defined by f(x) = 3x - 3 Prove or disprove: fis an isomorphism of the additive group…

A: The objective is to prove or disprove :f is an isomorphism of the additive group onto itself. Group…

Q: polynomial

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Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group

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Q: Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2 × G1.

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Q: Show that if G and H are isomorphic group, then G commutative implies H is commutative also.

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

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Q: Theorem Let f: G→ H be a group homomorphism. Then, Im f ≤ H. واجب Proof

A: A non empty subset of group is said to subgroup of group if it is also a group. We know that One…

Q: This is abstract algebra:   Prove that if "a" is the only elemnt of order 2 in a group, then "a"…

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Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

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: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

Q: (iv) Does there exist a group G such that [G, G] is non-abelian? Give an example, or prove that such…

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Q: Let (G1, ) and (G2, *) be two groups and p: G1 G2 be an isomorphism. The

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

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Q: 1,) and (G2,*) be two groups and →G2 be an isomorphism. Then *

A: given that G1,. and G2,*are two groups  and φ:G1→G2  be an isomorphism

Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

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

A: Here we use basic definitions of Group Theory .

Q: If a group G is isomorphic to H, prove that Aut(G) is isomorphic toAut(H)

A: We have to prove, If a group is isomorphic to H, then Aut(G) is isomorphic to Aut(H).

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

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