12. If a group G has exactly one subgroup H of order k, prove that H is normal in G.

Q: 16. If H is a subgroup of G, and K is a normal subgroup of G, prove that HK = KH. 1

A: Proof:Let H be a subgroup of G and K be a normal subgroup of G.We will proof HK=KHLet, hk be an…

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: 3. Consider the group S. a) For the subgroup H= {(1),(13)} , write all cosets that can be formed. b)…

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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: 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: 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: Suppose H is a distant and normal subgroup of a group G. Prove that each subgroup of H is a normal…

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

Q: Let N be a normal subgroup of a finite group G.  Prove that the order of the group element gN in G/N…

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Q: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

Q: 27. Let H be a subgroup of a group G and let g € G. Define a one-to-one map of H onto Hg. Prove that…

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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: 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: Let H be a subgroup of a group G with the property that for all a andb in G, aHbH = abH. Prove that…

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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: 4. Let G be a group of order 49. Prove that G is cyclic or g7 = e for all g € G.

A: Given a group of order 49, then to prove that either G is cyclic or g7=e for all g∈G.

Q: Let G = Sn. Assume that H is a subgroup of G, that H contains atleast one odd permutation, and that…

A: I have provided a detailed answer in the Explanation section.Explanation:This is a fascinating…

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: Let G be a group acting on a set S. Let x and y be elements in S such is in the orbit of x under the…

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Q: 12. Let : G→ H be a group homomorphism. suppose that K is a normal subgroup of G. Prove or disprove:…

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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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