Question 3: Leb 3:6>G' be an isomorphism . Prove the follawing assertions : la) IGl =1G| (b) lal =I gcall for every a EG. %D (6) G is Abelian iff G is Abelian - (d) G is cydic iff G' is cyclic .

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

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Q: Let R be set of Peal numbed Let Coeffa ] a cR fefa be i) Show G under emultiplich is abelian Sooup…

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Q: 3. (a) (i) Let p: (Z, +) → (C – {0}, :) be given by (n) = (-i)", for each n e Z. Prove or disprove…

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Q: 16. Let R* be the multiplicative group of non zero real numbers and let S= the group under the…

A: As per the guidelines I have solved first problem.

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

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Q: If G is non-Abelian, show that Aut(G) is not cyclic.

A: Given: G is non-abelian. To Show: Aut(G) is not cyclic.

Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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…

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Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

Q: 6. Use the Fundamental Homomorphism Theorem to prove that the following groups are isomorphic. (a)…

A: (a) Consider the function           f : Z3 × Z3 → Z3  defined by      f(a, b) = a - b   Then, for…

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e a^4 =…

A:

Q: Let φ : G → H be an onto homomorphism.(a) Assume that G is abelian. Does this imply that H is…

A: Let's analyze each part of the question: (a)  Assume φ: G → H is an onto homomorphism and G is…

Q: 51. Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove…

A: According to our guidelines we can answer only first question and rest can be reposted. Not more…

Q: E XAMPIE 2:- la, b, cs 5. (S 7 d X PCS) > P (8) A AB %3D abelien

A:

Q: 3 Let f: (A, SA)(8, 5,) be an order isomorphism and, Sa={xEA : x<ga}+ and Spa) -ly e8: f(5.)=Spa)…

A: Given: f: (A,⩽A) → (B,⩽B) is an order isomorphism Sa={x∈A: x&lt;Aa}≠ϕ Sf(a)={y∈B: y&lt;Bf(a)} To…

Q: 3. Are the rings 2Z, 3Z isomorphic. Any isomorphism is linear for addi- tion, so it has the form f…

A: Ring Isomorphism: Letus consider two rings R and S.Let us consider the mapping  f: R→S. Then it is…

Q: Chapter 2, exercise 5.1: Let : G → G' be a surjective homomorphism. Prove that if G is cyclic, then…

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Q: 14. If G is abelian and p: G→ G' is a homomorphism of G onto G', prove that G' is abelian.

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Q: Let f. → (R, +> be defined by f(x) = 3x- 3 Prove or disprove that f is an isomorphism from the…

A: This is related to isomorphism

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Q: 5. Find the kernels and images of the following homomorphisms. Which of the homomorphisms are…

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Q: Consider the following permutations in S-: 2 3 4 2 3 4 5 6 BG 3 2 7 15 6 = (! a = 1 4 3. 4 their…

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Q: ) Given the set S = {1,2,...,n}, show that Sym(S) is Abelian for n = 1, 2, and non- Abelian for n>2,…

A: This is a problem with the theory of Groups.

Q: Find a rule for the following isomorphism: 9: Z2 x Z3 + Zz x Z2 p (a, b) = (

A: we have φ:z2×z3→z3×z2 we know a result: A G1⊕G2 is isomorphic to G2⊕G1 by g1,g2→g2,g1.

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Q: 12. (Exercise 16.22) Prove the third isomorphism theorem. At each major step in the proof, explain…

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Q: iv Sketch the Caley Graph of the additive Group of direct product Z3× Z4 with respect to the…

A: Consider the conditions given in the question. Clearly from the hint a torus is involved in the…

Q: Show that any automorphism of GF( pn) acts as the identity onGF( p).

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Q: Prove that Sn is not abelian for all n > 3.

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Q: 5. Let G be a group and define z" z * *** *z for n factors c, where z E G and n E Z+. (a) Suppose…

A: Given that G is a group. Also zn is defined as follows. zn=z*z*z*...*z (a) Use mathematical…