Let G be a group and H a subgroup of G. H is a normal (or distinguished) subgroup of G if and only if (Va € G)(aH = Ha), where aH = {ah | h H} and Ha= {hah € H} respectively. Recalling that the sets aH and Ha are called cosets of H, this definition says that H is normal if and only if the left and right cosets corresponding to each element are equal. We will meet cosets again when we pick up our reading of Hölder in the next section. The tasks in the rest of this section first provide some practice with using Definition l' and two other methods that can be used to prove a particular subgroup is normal. Task 5 Let G be a group, and recall that the center of G is the subgroup defined by C = {rG| (Vy € G) (yx = xy)}. Use Definition 1' to prove that C is a normal subgroup in G. (You can assume C is a suboroun of G and inst prove the normality of Cin G)

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

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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

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Q: Z, x is not a group.

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Q: 2. Let G = Z50 and consider it's subgroup H = (5). Find all coset representatives of 3+ H.

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Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

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Q: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

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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: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

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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: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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Q: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

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Q: Write U(32) as the internal direct product of two proper subgroups.

A: Given:         U32 We use the theorem namely Ut≈Usst   because Usst is a subgroup of Ust

Q: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

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Q: Let G and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

Q: 3. Let G be a group, HAG, KAG,HnK = {e}. Show that G= G/H x G/K.

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Q: Let H be a subgroup of G and let a, be G. If Ha Hb, then* %3D aH = bH O a-1H = b-1H O Ha = Hb Ha-1 =…

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Q: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

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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…

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