(b) Find all bEG with order /b| = 7 bEG. are: (c) Construct in Ga subgroup H with index.(H) = 7

Q: If p is a prime, prove that any group G of order 2p has a normal subgroup of order p and a normal…

A: To prove that any group of order 2p has a normal subgroup of order p and a normal subgroup in g

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

A: According to the given information, it is required to find a non-trivial normal subgroup of the…

Q: Prove that a group of order 375 has a subgroup of order 15.

A:

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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Q: 2) Let H be a normal subgroup of G. If| H|-2. Prove that H is contained in the center Z(G) of G.

A:

Q: Show that the translations are a normal subgroup of the affine group.

A: To show: The translations are a normal subgroup of the affine group

Q: 11. Find the cyclic subgroup of D4 generated by µp². What is the order of this subgroup?

A:

Q: Show that the center of a group of order 60 cannot have order 4.

A:

Q: Determine the subgroup lattice for Z12.

A: By fundamental theorem of cyclic group, Subgroups of cyclic group ℤn are again cyclic and are…

Q: (2) of order 5 is in H. Let G be a group of order 100 that has a subgroup H of order 25. Prove that…

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Q: 3) a) Explain how many distinct necklaces of 11 red and green beads are possible? b) Prove that a…

A: Note: As per Bartleby guidelines, for more than 2 different questions asked, only 1 has to be…

Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: Draw the subgroup lattice for Z28-

A: Draw the subgroup lattice for Z28

Q: In Z, find all generators of the subgroup <3>. If a has infinite order,find all generators of…

A: Since a has infinite order, the same holds for a 3 since if it would have order n < ∞, then 1 =…

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: Find all subgroups of S and its subgroup graph.

A: We will use basic knowledge of groups and abstract algebra to answer this question correctly and in…

Q: Find all the subgroups of Z48. Then draw its lattice of subgroups diagram.

A:

Q: 17. Show that every group of order (35)° has a normal subgroup of order 125.

A:

Q: Describe all the elements in the cyclic subgroup of GL(2,R) generated by the given 2 × 2 matrix. -1

A: Let A=0-1-10∈GL2, R Then the cyclic subgroup generated by A denoted by A=An| n∈Z

Q: Prove that every subgroup of the quaternion group Q8 is normal. Deter- mine all the quotient groups.

A:

Q: Show that Z has infinitely many subgroups isomorphic to Z.

A: We have to show that Z has an infinitely many subgroups isomorphic to Z.

Q: In Z24, list all generators for the subgroup of order 8. Let G = &lt;a&gt;and let |a| = 24. List all…

A:

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: Find the three Sylow 2-subgroups of S4

A:

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: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: ake the minimum three groups along with subgroups (Mathematically) and verify the langrage theorem

A: Lagrangian theorem says that, If H is a subgroup of a group G then H\G. Now take, Z3={0,1,2} The…

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

A:

Q: Show that every group of order (35)3 has a normal subgroup of order 125

A: Given, A Sylow 5-subgroup of a group of order 353 is of order 125. The divisors of 353 that are not…

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: Please help me understand the following question and please explain the steps. Picture is below

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

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Q: Draw the lattice of the subgroups Z/20Z.

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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: Show that there are two Abelian groups of order 108 that haveexactly one subgroup of order 3.

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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: Let |G| = 15. If G has only one subgroup of order 3 and only one oforder 5, prove that G is cyclic.…

A: Note that for a non identity element a ∈ G, , |a| =3,5, or 15. Now let A= {a ∈ G | |a| = 3} and B =…

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

A:

Q: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

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: (a) Let G be a non-cyclic group of order 121. How many subgroups does G have? Why? (b) Can you…

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Q: Prove that a group of order 595 has a normal Sylow 17-subgroup.

A:

Q: 6. If N< G and G/N is free, prove that there is a subgroup H such that G = HN and HoN= 1. (Use the…

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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) Find all subgroupsof (Zs.+s).

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Prove that every group of order 375 has a subgroup of order 15.

A: According to the given information it is required to prove that every group of order 375 has a…

Q: Suppose that H is a subgroup of Sn of odd order. Prove that H is asubgroup of An.

A: Given: H is a subgroup of Sn of odd order, To prove: H is a subgroup of An,