Exercise 6.5. Let G be a group and let S := {abab | a, b G}. The commutator subgroup of G is the subgroup C generated by S, i.e. C = Ɑ H. (1) Show that C is a normal subgroup of G. H:H

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: 2.26. If G is a group and H is a subgroup of index 2 in G, then prove that H is a normal subgroup of…

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Q: 15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk =…

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Q: Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the…

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Q: Let G be a group and let a be a fixed element of G. Show that H = {x E G |xa = ax} is a subgroup of…

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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: 10.3.3. Let G Z/4Z x Z/6Z, and let H be the subgroup of G generated by (2, 2). (a) What are the…

A: Given that G=ℤ/4ℤ×ℤ/6ℤ.i.e. G=ℤ4×ℤ6H is a subgroup of G generated by 2,2i.e. H=<2,2> oG=4×6=24

Q: (5) Let G be a group. Let H be a subgroup of G and K be a normal subgroup of G. Prove that HK is a…

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Q: 2. Suppose an element a E G has order 2. Prove that (a) is a normal subgroup of G if and only if a…

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Q: 4. Let G be a group and G' be the subgroup generated by the set S = {x¹y¹xy x, y = G}. G' is called…

A: Let G be a group, G' be its commutator and N be a normal subgroup of G such that G/N is abelian.…

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

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Q: 10. Let G be abelian and let H be a subgroup of G. Show that G/H is abelian.

A: We have to prove that given theorem.

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: Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с.…

A: We have to find order of subgroup of G generated by 2.

Q: 2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G ×…

A: Step 1: Step 2: Step 3: Step 4:

Q: 5.3 Let H be the set of elements () of GL(2, R) such that ad-bc-1. Show that H is a subgroup of…

A: be the set of elements  of  such that  that is the We have to show that  is a subgroup of

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

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Q: Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e,…

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Q: 1. Let G₁ and G₂ be two groups, with identities e₁ and e2, respectively. Let G = G₁ x G2 and let H…

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Q: QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a E G} and we…

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Q: Let K be a subgroup of G and let H = {g : gKg^-1} = K. Show that H is a subgroup of G

A: Let K be a subgroup of G and H be defined as H = { g : gKg-1 = K }. Then, we have to show that H is…

Q: 1. Let G be a group. Define the center of G by C(G) = {x € G: xg = Prove that C(G) is always a…

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Q: ¹) Let H be a subgroup of a group G. Define NĠ (H) : = {g €G | g Hg¯¹ = H}. Prove that N(H) is a…

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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: Definition 4.24: Let G be a group, and let Ω be the set of subgroups of G. Let G act on Ω by…

A: We prove our claim by using definition of NG(H).

Q: Theorem(7.9): If (H, *) is a subgroup of the group (G, *). then Va e G the pair (a+H a,+) is a…

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Q: 2- Find the cyclic subgroups in (Z,, +,) generated by 3,5.

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Q: 3. Let A and B be groups. Show that S := {(a, e) | a € A} is a normal subgroup of G:= A x B and the…

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Q: Let G = U56 and define two subgroups H₁ and H₂ of G by := H₁ := Calculate H₁ H₂|. {1, 5, 9, 13, 25,…

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Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

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Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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Q: 2.1.12. Corollary Let (H)be a nomal subgroup of the group (G). There is one -one correspondence…

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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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Q: 2. Let G be a group and let H be a subgroup of G. Define N(H) = { x = G | xHx™¹ = H}. Prove that…

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Q: Let G and H be two groups, , & € Hom (G, H), and A C G be a nonempty subset. (a) Prove ((A)) = (…

A: We have proved (a) in two steps, showing both the subset relation one by one. For (b) we have used…

Q: Let G be a group. If a G define C(a) = {g €G: gag¹=a}. Let a € G. Prove the following (a) Prove that…

A: The main objective is to prove the given statements.

Q: Find the partition of Ze into cosets of the subgroup H = {0,3}.

A: In this problem we have to find the partition of  into cosets of the subgroup .

Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…

A: Given G=S3 and A=1,1 2 3, 1 3 2 The objective is to show that CGA=A, NGA=G. The definition for…

Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | |   2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…