4. Let G be a group of order 14. Using Lagrange's theorem, show that every subgroup of G either contains an element of order 2, or contains an element of order 7, or is the trivial subgroup {1}. Now show that if a subgroup contains an element of order 2 and an element of order 7 then that subgroup is equal to G.

Q: Exercise 3.2. Let G, G' be two groups and : G→G' be a (group) homomorphism. (1) Given a subgroup H…

A: Solution-

Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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Q: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

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Q: Prove the next statement: If H is a subgroup of a finite abelian group G, then G has a group…

A: Introduction: In mathematics, an isomorphism is a type of mapping or function between two…

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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Q: If H and K are subgroups of G and g belongs to G, show that g(H N K) = gH n gK.

A: Given: H and K are subgroups of G and g.To prove: gH∩K=gH∩gK

Q: (3) Let H be a subgroup of a group G. Let a E G. Prove that a¹Ha = {a¹ha he H} is a subgroup of G.

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Q: Prove that any non-trivial subgroup of (Z, +) is isomorphic to (Z, +).

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Q: Let G be a group of order 30. Does G have to have a subgroup of order15? Does G have to have an…

A: Group of order 30 does not necessarily have a subgroup of order 15, nor does it necessarily have an…

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

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Q: Let a e G. Prove that $(a") = ¢(a)" for all n e Z.

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Q: 9. Let G and H be two groups, and G = {(9, en) | g = G}. Prove that G is a normal subgroup of G x H,…

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Q: 18. Let G be the group (Z/7Z)x. Determine all the cyclic subgroups of G.

A: In this context,  represents the multiplicative group of integers modulo 7 that are relatively…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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Q: 6. Let G be GL(2, R), the general linear group of order 2 over R under multiplication. List the…

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Q: Question #1: Let G = Z20 , H = be the subgroup of G generated by 5, and K = be the subgroup of G…

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Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

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