7. [10] Let G be a group, with x € G and o (x) = 48. Give the order of the following elements of G: %3D а) о(x?) b) o(x20) c) o(x23)

Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

Q: For G being each of the following groups, compute the centre Z(G): (a) S3 (b)Z/3Z x S3

A:

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: Let G be a group of order 27. Let x = G. Prove that x³ € Z(G).

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Q: If U(14) = , then U(14) is cyclic group generator by a) 5 b) 11 c) 4 d) None of the above

A: We have to solve given problem:

Q: 8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set…

A:

Q: Show that (Q*,-) is not cyclic..

A: We use the basic definitions of cyclic groups to solve this problem. The detailed solution is…

Q: Let A = 314⁰ 3 2 NO CON 1 -3 OOTO (a) M12 and C12 (b) M23 and C23 0 0 6 2 -1 -4 7 0 0 0 Find the…

A: Since you have posted a question with multiple subparts, we are supposed to answer only the first…

Q: How would you solve such a question: Suppose f : Z → Z5 is a group homomorphism, and suppose f (3) =…

A:

Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

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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: Exercise 6.3.12. Suppose G is a group, a, b e G such that gcd(la\, [b|) = 1. Prove (a) n (b) = {e}.

A: Suppose (G,.) Is a group . Let , order of a and b is m and n respectively. Then am =e , bn =e .…

Q: 4.1.7. Let G = Dg act on N = D8 by conjugation. Make an 8 × 8 table, where the rows are indexed by…

A: It is given that G=D8 act on Ω=D8 by conjugate. So it follows that, D8=e, r, r2, r3, s,sr, sr2,…

Q: 2. Which of the following are groups? For those which are not groups, explain why not (it is enough…

A: 2)(i)  under   is  not  a  group  since which has  no  inverse  under    (ii) Let      Then,   is…

Q: 42. Let G be the group of 2 × 2 matrices under addition and b •{(a $): a +d=0}. d H = Prove that H…

A:

Q: . Let G be a group, let g e G, and let H - G. Suppose that the element Hg E G/H has order n. Show…

A: We have to prove that given statement:

Q: (a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove…

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Q: 1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h)…

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Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________

Q: 4: Find possible orders of elements in the groups S3, S4, and S5.

A: