10 2) (i) Prove that H = (0, 2, 4, 6, 8) is a subgroup of Z (ii) Prove that H is a normal subgroup of Z (iii) Compute the left cosets of H in Zo (iv) Describe the factor group (also called the coset group) Z/H by giving a well-known group to which it is isomorphic. Explicitly, give the isomorphism. 10

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Q: 2. First Isomorphism Theorem. Let G be a group and H-((.s)|ge G). (a) (b) (1) G is abelian. (i) H is…

A: Given G is a group then, G×G is a group of the type such that G×G={(a,b); where a, b belong to G}…

Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

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Q: Let (G, ') denote the set of all 2 x 2 real matrices A with det{. and det {A} € Q (the rational…

A: a)  Let A, B ∈ ( G, ·) ⇒ det {A} ≠ 0, det {A} ∈ QAlso det {B} ≠ 0, det {B} ∈ QNow det {AB}= det {A}…

Q: 22. Let 0 G → H be a group homomorphism, and let K = ker = {a € G|0(a) = EH}. Show that gK = Kg for…

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Q: (b) Let H= ((3,3, 6)), the cyclic subgroup of G generated by (3,3,6). Determine |G/H.

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Q: 1. In the Commut-tive group (G),define the set H by H=lacG|=e for Some KE Z} Prove that (H,*) is a…

A: According to our guidelines, we can answer the first question and rest can be reposted.

Q: (5) Let G be a fnite akelian дроиер Suppen the G=E8₁, 8₁, ---, gn) G. of oxter n. Supp Xe Gi (a)…

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Q: is a subgroup of Z1, of order: 3 12 O 1 The following is a Cayley table for a group G. 2. 3.4 = 2 3…

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Q: Let G = : a – b = c – d, a,b, c, d E R Show that G is a group under (the usual) matrix addition.

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A: (Solving the first question as per our guidelines) Question: Show that S3 is meta-Abelian.

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Q: QUESTION 6 Consider the groups U(8) and Za (i) Determine the identity element in the group U(8) × Z.…

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Q: t the groups: G, G', G'' and the functions f:G-->G', g:G'-->G'' Prove the following statement: If…

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Q: (6) Let G be a noncommutative group. Show that G has a nontrivial subgroup.

A: We recal Cauchy's theorem and then use it

Q: QUESTION 6 Consider the groups U(8) and Z () Determine the identity element in the group U(8) x ZA…

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Q: Suppose that f:G - G such that f(x) = axa. Then fis a group homomorphism if and only if a^4 = e O…

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Q: the product of the elements is the identity. 7 has order 2. of order 2.

A: Let us prove above result

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Q: -1 а Let G = : a € Z4 or 0 1 Construct the multiplication table of G (under the usual matrix…

A: Given G=1a01 or -1a01: a∈ℤ4. Hence G has total eight elements. Therefore multiplication table of G…

Q: Qi/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*) Q2/ In (Z9, +9)…

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