3. Show that if n ≥ 5, then the only normal subgroups of Sn are {}, An, and Sn. You may use the theorem that An is simple for any n ≥ 5. (Hint: Let HSn. Show that (HnAn) ◄ An, so HnAn = {e} or An. If HnAn An, show that H An or Sn. If HnAn = {ɛ}, use the formula |HA| = |H| · |An\/|H^ An to show that |H| ≤ 2.) = =

Q: 8. Find a non-trivial normal subgroup of the octic group. Demonstrate that this subgroup is normal.

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Q: Let (a1a2a3a4) and (a5a6) be disjoint cycles in S10. Show that there isno element x in S10 such that…

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Q: 3. Prove that if d is a common divisor of a and b, then d = gcd(a, b) if and only ir gcd(a/d, b/d) =…

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Q: 13. Which of the following is a subgroup of (R+, *) where a*b = (ab)/2? (Q, *) A. B. (Z, *) C. (Q+,…

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Q: 2. LetG = (Z12,) & H == {4,8,0}. Then [G:H]=3. *True *False

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Q: Determine whether the intersection of normal subgroups is a normal subgroup. Show your work.

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Q: 18. With H and K as in Exercise 16, prove that HMK is a normal subgroup of H.

A: Please complete the missing important information for further explanations.

Q: how that in D4 is not a normal subgroup. Please explain in details and show all the works.

A: To Show, a subgroup in D4 is not normal.

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Q: 2. Let X have the each of the following cdfs. For each case, find the cdf of Y = X². a. 22 Fx(x) = 0…

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A: Topic = Algebra

Q: Question 7. (10 Marks) If K is a subgroup of G and N is a normal subgroup of G, prove that KnN is a…

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Q: Let N be a normal subgroup of A4 containing a 3-cycle. Show that N = A4.

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Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: n ΔMNO MO is extended through point O to point P, m∠NOP=(7x−17)∘, m∠MNO=(x+20)∘, and m∠OMN=(3x−7)∘.…

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Q: 23. Every element in Z4 Z8 has order 8. . TRUE ● FALSE An is a normal subgroup of Sn. TRUE FALSE 24.…

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Q: Let Dn = {1, a, ...‚ an-¹, b, ba, .. - ,ban-1} with o(a)= n, o(b) = 2, and aba = b. (a) Show that…

A: Let's first establish some properties of the group Dn:1. The group Dn is generated by a and b.2. The…