1. Let p e Z be a prime number and set Z, = {- e Q: If gcd(n, m) = 1, then p Im}. %3D a. Show that Z, 4 Q. b. Let Z(p) be the quotient group Q/Zp. Show that Z(p) is an infinite group. Hint: Show that Z(p) contains an infinite subset.) c. Recall that a p-group is a group whose elements all have order some power of p. Show that the quotient group Z(p) is a p-group.

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

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Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

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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: t Z(p®) is an infinite gr

A: Here, Un =<1pn  +ℤp >∀n∈ℤ+ So, Uo =e,U1 =<1p +ℤp >,U2 =<1p2 +ℤp ,U3 =<1p3 +ℤp…

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: Give the lattice of subgroups of D₁.

A: This question from group theory , related to topic lattice of subgroup.

Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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Q: Prove that An even permutation is group w.y.t compostin Compostin function.

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Q: let n be a fixed natural number.Verify that the set  n.Z={n.k l kEZ} is a group of (Z,+).

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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: Q17 : If a and b are elements in a group G, then the order of ab is the same as the order of ba.…

A: Let us take a non abelian group then their elements will never commute  that is a*b not equal to b*a…

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A: Detailed solution attached in step 2.

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

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Q: 4.14 Give an example of an infinite group G such that every element of G has finite order. 4.15 a)…

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Q: disprove: If G is a group and H, K < G are two subgroups, then (HK)≤ G.

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