6. Find Sylow subgroups of a group G of order 108. Show that either G has a normal Sylow 3- subgroup or G contains a normal subgroup of order 9.

Q: compute the 3 -sylow subgroups of S5

A: n3 (S5 ) =(1+3k) /40

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: Suppose that H is a subgroup of Z under addition and that H contains 250 and 350.  What are the…

A: To determine the possible subgroups H satisfying the given conditions

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

A:

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

A:

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: Find a noncyclic subgroup of order 4 in U(40).

A: Let U(40) be a group. Definition of U(n): The set U(n) is set of all positive integer less than n…

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 H is a subgroup of Z under addition and that H contains250 and 350. What are the…

A:

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

A:

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: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.

A:

Q: Suppose that H is a proper subgroup of Z under addition and that Hcontains 12, 30, and 54. What are…

A: It is given that H is a proper subgroup of Z under addition and that H contains 12, 30 and 54.

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 finite group and let primes p and q ≠ p divide |G|. Prove that if G has precisely one…

A: Given that G be a finite group. Consider G be a finite group and consider prime p and q such that…

Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is

A: It s given that H and K are subgroups of G, H=18 and K=30. Since H, K are subgroups, H∩K≤H  and…

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

A:

Q: 4. If H is a subgroup of G, then show that the set W = ngHg¹ is a normal 9€G subgroup of G.

A: Given That : H is a subgroup of G  To Show: The set W=∩g∈GgHg-1 is a normal Subgroup.

Q: 1. Prove that a subgroup which is generated by W-marginal subgroups is itself W-marginal.

A:

Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: 15. Suppose that N and M are two normal subgroups of a group G and that NO M = {e}. Show that for…

A:

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

A:

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

A:

Q: Find the three Sylow 2-subgroups of S4

A:

Q: The subgroup {e} is called the nontrivial, that is, a subgroup that is not e is nontrivial.…

A:

Q: Q7. Suppose that the index of the subgroup H in G is two. If a and b are not in H, then ab ∈ H.…

A:

Q: If H and K are normal subgroups of G, show that their intersection is also a normal subgroup. To do…

A:

Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

Q: a) S3; 6) A4.

A: (a) Given group is S3. Since order of group is S=6=2×3. Therefore number of Sylow 2- subgroups are:…

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: 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: Prove that a normal subgroup need not to be a characteristic subgroup.

A:

Q: If H and K are subgroups of G, Show tht H intersecting with K is a subgroup of G.  (Can you see that…

A: Use the 2-step subgroup test to prove H Ո K is a subgroup, which states that,

Q: If G is a group with 8 elements in it, and H is a subgroup of G with 2 elements, then the index…

A: We are provided that a group G with 8 elements and H is a subgroup of G with 2 elements and…

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: 2. A Sylow 3-subgroup of a group of order 54 has order

A:

Q: What is the relationship between a Sylow 2-subgroup of S4 and the symmetry group of the square? that…

A:

Q: 189. Let be given Ga finite group and Pe Syl,(G). Give an example of a subgroup H of G where HnP is…

A:

Q: H. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.

A:

Q: Prove that every group of order 675 has a normal Sylow 5-subgroup.

A: Theorem: If number of Sylow p-subgroup of a group G is unique, then that subgroup is normal. Let G…

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

A:

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: Since 11 is an element of the group U(100); it generates a cyclic subgroup Given that 11 has order…

A:

Q: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

Q: 5. Show that the intersection of two normal subgroups of G is a normal subgroup of G 6. If G is a…

A:

Q: 1- (Z,+) is a subgroup of (Q,+) 2- (Q,+) is a a subgroup of (R, +) 3- (R,+) is a a subgroup of (C,+)…

A:

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

A:

Q: Suppose that | G|= 20. What're possible orders for a subgroup of G. 35.

A: We can find the subgroup of G as below.