6. For a positive integer n 2 4 and a prime number psn, let U, denote the union of p,n all p-Sylow subgroups of the alternating group A, on n letters. Also let K, P, n denote the subgroup of A, generated by U. and let p, n Kpn denote the order of Kn. Then P, n . (a) K2,4| = 12 (b) |K2 4| = 4 (c) K2.5 = 60 (d) K3.5 = 30 \K3,5

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

Q: When do you use a t-test for independent groups and what do the numerator and deniminator in the…

A:

Q: Sylow-5 subgroups-

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Q: 2n 5) Let G be a group such that |G| = (xd,)!, and |H|= (n – 1), where H a Subgroup of G, then- n(n…

A:

Q: Prove or disprove that H = {nEZ\n is divisible by both 8 and 10} is a subgroup of Z. What happens if…

A: Given: H=n∈Z n is divisible by both 8 and 10We need to show whether H is a subgroup of Z or not and…

Q: Without using the 2 . Odd Test, prove that there is no simple groupof order 210.

A: In this question, we have to show that there is no simple group of order 210.

Q: Let G be group, aeG ond lal=n. Show that laml = (m,n) %3D

A:

Q: Consider from the b the Symmetric group. 1 2 3 5 6 7 7 5 4 1 2 36 3 5 6 = 3 7 15 a) Express each of…

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Q: 9. Let G be a cyclic group with |G| = 24 and let a be an element of G. If a - cidentity) and a" -…

A: Let G be a group of order 24 and let a∈G. Given a8≠e and a12≠e. Prove G=<a>.

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

A: Given that                    ℝ+, * is a group             where…

Q: Check whether the following are groups. Justify your answer. (i) (G, *) where G= {1, 2, 4, 8}, and *…

A: To check G is a group under the given operations * and ∇. G,· is a group only if it satisfies the…

Q: 3. Suppose that a group G contains elements of orders 1, 2, 3, 4, ..., 8, 9. What is the minimum…

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Q: Find all subgroups of U(7)

A:

Q: Every cyclic group is a non-abelian group.

A: False, all cyclic groups are abelian group. This is true

Q: 5. If H = 122Z and K = 8Z are subgroups of (Z, +). Then H + K = ... %3D

A:

Q: The elements of the quotient group (2/8).) are {} 2- -208,-108,008,108,208,...) 3- (0,1,2,3,...,8)…

A: Given that the quotient group Z8.⊗. We have to find the elements of the quotient group Z8.⊗.…

Q: What is the difference between a proper subgroup and a subgroup?

A: Here, we will tell the difference between a subgroup and a proper subgroup.Order of group means…

Q: Subsets, improper subgroup, trivial subgroup and non-trivial proper subgroup of Z4. b. Subsets,…

A:

Q: Depine Dihedral group of order 8. Write two non-trival Subgroups of Dihedral group

A: Dihedral group of order 8 is given by     D4 = group of symmetries of a square          = {R0, R90,…

Q: Find the order of the given factor group (Z3 x Z6)/

A: If G be a group, and H be a subgroup of G, then the order of the factor group G/H is given by…

Q: show that the set Z ={0,+-1, +-2,...} with the operation of multiplication (a,b) map to ab is not a…

A: As per the bartleby guideline I can solve only 1st question or 3 sub questions of 1st question if…

Q: 4*. Let a and b be elements of orders 5 and 8 in a group of order 120. List the possible orders of…

A: To list the possible orders of the subgroup generated by a and b under the given conditions

Q: 7. (a) What is the order of (1, (123)) in Z4 × S3 ? (b) Write down all the distinct cosets of the…

A:

Q: 5. How many automorphisms does Klein's 4-group have?

A: No of automorphism of Klein's 4-group:  K4={e,a,b,c} f1=eabceabc = I f2=eabceacb = bc f3=eabcecba =…

Q: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

Q: 9. Find all proper nontrivial subgroups of Z₂ × 2₂.

A: The given group is: .To Do: We have to find all proper nontrivial subgroups of .Note: Since you have…

Q: Alte

A:

Q: 2. What is the cyclic subgroup of Z generated by -1 under +?

A: Let, H = <-1> Now, for any positive integer k,       -k = (-1) + (-1) + (-1) + . . . . . . . .…

Q: Are there any other subgroups of order 3 & 2 ? For his vere'll consider all subsets of 3 and 2…

A: We see that S3 is a group of six elements. f is the element in S of order 3. Hence, <f>, <f…

Q: 2- Find the cyclic subgroups in (Z,, +,) generated by 3,5.

A:

Q: EQ6/ Give example of a group such that has element of order 7 and nurmal subgruup of Order 3.

A: A detailed solution is given below.

Q: The n-th dihedral group Dn is the group of symmetries on a regular n-gon. The group of symmetries on…

A: Given : The n-th dihedral group Dn is the group of symmetries on a regular n-gon. The group of…

Q: Q2 / If a, b and c are elements of a group (G,*) such that c*a=c*b then a=b.

A:

Q: 2. Is the set Z3 = {0,1,2} form a group with respect to addition modulo 3 how about to…

A: We use caley's table to verify the properties of a group.

Q: The elements of the quotient group (2/8). 12 (0,1,2,3,...,8) 1- 2- -208,-108,008,108,208,...) 3- 4-…

A:

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

A:

Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | |   2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…