Suppose that G is cyclic and G = (a) where Ja| = 20. How many subgroups does G have?

Q: Suppose that a group G of order 231 has a normal subgroup N of order 11. Then, G/N is cyclic O False…

A: Given that G is a group of order 231 and N is an normal sub-group of G of order 11. To show: G/N is…

Q: How many cyclic subgroups does U(15) have?

A: To find Number of  cyclic subgroups does U(15) have

Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

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Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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Q: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

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Q: Draw the subgroup lattice for Z28-

A: Draw the subgroup lattice for Z28

Q: O In a group (G,x), if IGl= 37, the number of Passible Subgroups in G are

A: Note: Since you haven't mentioned which question you would like to get answered. We are providing…

Q: Suppose that K is a proper subgroup of H and H is proper subgroup of G. If |K| = 42 and |G| = 420,…

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Q: Suppose that H is a subgroup of a group G and |H| = 10. If abelongs to G and a6 belongs to H, what…

A: Given: H is a subgroup of a group G and |H| = 10 To find: If a belongs to G and a6 belongs to H,…

Q: Let Z denote the group of integers under addition. Is every subgroup of Z cyclic? Why? Describe all…

A: Yes , every subgroup of z is cyclic

Q: H be a subgroup of G.

A: We have to find out the truth value of the given statements. It is given that H is a subgroup of G.…

Q: Let G be a cyclic group of order n. Let m < n be a positive integer. How many subgroups of order m…

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Q: If a cyclic group T of G is normal in G; then show t subgroup of T is a normal subgroup in G

A: Given: A cyclic group T of G is normal in G.

Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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Q: If H and K are two subgroups of finite indices in G, then show that H ∩ K is also of finite index in…

A: If H and K are two subgroups of finite indices in G, then show that H ∩ K isalso of finite index in…

Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…

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Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

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Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).

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Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

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Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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Q: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

Q: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

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Q: Find all of the distinct subgroups of Z90 and draw its subgroup diagram.

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Q: (c) Prove that for every divisor d of n, Zn has a unique subgroup of order d. (Hint: What is the…

A: C) Let k be a subgroup of order d, then k is cyclic and generated by an element of order k =K⊂H…

Q: let H be a normal subgroup of G and let a belong to G. if th element aH has order 3 in the group G/H…

A: H is normal subgroup of G. And a belongs to G.  O( aH) = 3 in G/H and O(aH) in G/H divides  O(a) in…

Q: 3. How many cyclic subgroups does S3 have?

A: The objective is to find the number of cyclic subgroups of S3. Subgroups of S3 are, H1=IH2=I, 1…

Q: Show that every subgroup H of the group G of index two is normal.

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Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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Q: The group (Z6,6) contains only 4 subgroups

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Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

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Q: How many subgroups does Z/60Z have?

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Q: (a) Compute the list of subgroups of the group Z/45Z and draw the lattice of subgroups. (prove that…

A: In the given question we have to write all the subgroup of the group ℤ45ℤ and also draw the the…

Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…

A: The identity element in a subgroup H of a group G must be the same as the identity element in G.

Q: Prove that the intersection of two subgroups is always a subgroup.

A: In this question, we prove the intersection of the two subgroup of G is also the subgroup of G.

Q: Find all cosets of the subgroup of Z12.

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Q: Let Z denote the group of integers under addtion. Is every subgroup of Z cyclic? Why? Describe all…

A: Solution

Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

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Q: Prove that any group of order 75 can have at most one subgroup of order 25.

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Q: The Kernal of any group homomorphism is normal subgroup True False

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Q: The group (Z6,+6) contains only 4 subgroups

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Q: Suppose that X and Y are subgroups of G if |X|=28 and |Y|=42, then what is

A: "According to Bartleby Guideline, Handwritten solution are not provided" Given,            |x|=28…

Q: Is every subgroup of Z cyclic? Why? Describe all the subgroups of Z.

A: A subset H of G is called a subgroup of G if H also form a group under the same operation.

Q: Let let G₁ be A be of Suppose Subgroup index a group and a normal of finite G+₁ that H

A: We know that if G is a group and H is a subgroup of G and x is an element in G of finite order n. If…

Q: Q2// Let Hi family of subgroups of (G, *). Prove that the intersection of Hi is also * .subgroup

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Q: 1. Let G = Z/36Z. (a) Find all subgroups of G, describe all containments between these subgroups,…

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