(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 subgroup of G.

Q: Let N, K be two normal subgroups of a group G. Show that NK is a normal subgroup of G.

A: Given that N and K be two normal subgroups of a group G Since N and K are normal subgroup of G…

Q: Let H be a normal subgroup of G and K a subgroup of G that containsH. Prove that K is normal in G if…

A: Assume that H is a normal subgroup of G, and that K is a subgroup of G containing H. Then, we need…

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: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

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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 be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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Q: Let G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

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Q: d) G=(Q-(0}, ); H=Q+ e) G=(Zg, ); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under…

A: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of…

Q: f f is a homomorphism of group G into a  group G' with Kernal K, then K is a normal  subgroup of G.

A: If f is a homomorphism of group G into a group G' with Kernlal K, then we have to prove that K is a…

Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

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Q: Let H and K be normal subgroups of a group G such at HCK, show that K/H is a normal subgroup of G/H.

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Q: Let G be a group and H a subgroup of G. If [G: H] = 2 then H ⊲ G, where [G: H] represents the index…

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Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

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: (4) Let G be a group. Prove that Z(G) is a normal subgroup of G.

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Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: The intersection of any two normal subgroup  of a group G is a normal subgroup of G.

A: We have to prove that the intersection of any two normal subgroup of a group G is a normal subgroup…

Q: Let G be a group, let H≤G be a subgroup, and let N G be a normal subgroup. (i) Show that HnN is a…

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Q: why is Q/Z(G) is a subgroup of G/Z(G)?

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Q: A group G of order 72 has a subgroup H of order 24. Prove that either H is normal in G or H has a…

A: To prove this statement, we can use the fact that if a group of order 72 has a subgroup of order 24,…

Q: 2. Let G = Z₂ x Z₁. (a) Find subgroups H and K of G such that HK but G/H/G/K. (b) Find subgroups A…

A: (a) Here, G = Z2 × Z4                = { (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1,…