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 elements of the quotient group (/18) are 1- 2- {.-208,-108,008,108,208,.} 3- (0,1,2,3, .8) 4-…

A:

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: Question 7. Prove or disprove: (a) The union of two subgroups of a group (G, *) is a subgroup of G.…

A: 7)  We need to prove or disprove the given statements. a) The union of two subgroups of a group G ,…

Q: Sylow-5 subgroups-

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Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A: Given that 2Z is the set of all even integers.

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: 23) Let the group ({0, 2, 4, 6, 8, 10), +12), find all distinct cosets of a proper subgroup H. Q4)…

A: We work with the abelian group Z_12

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: 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: (b) Let H= ((3,3, 6)), the cyclic subgroup of G generated by (3,3,6). Determine |G/H.

A:

Q: Find all subgroups of U(7)

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Q: If G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G with H…

A: Given: Group G=1,2,3,4,5,6 under multiplication modulo 7. Subgroup H=1,2,4 of G. To find:   a) Right…

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: Consider the subgroup (a) {4, 8, 12, 16} H = (6) of Z24. Which of the following equals 4 + H? (b)…

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Q: list the elements of the subgroup generated by the subet {2,3} of Z12

A: Let

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

A:

Q: 4. Consider the group A4, a group with twelve elements. Note that the positive divisors of 12 are 2,…

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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: 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: In Z24, find a generator for each of the following subgroups. (a) (21) n (10). (b) (15) n (14). (c)…

A:

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: 9. Let G and H be two groups, and G = {(9, en) | g = G}. Prove that G is a normal subgroup of G x H,…

A:

Q: If G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G with H…

A: Given G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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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: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

A:

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: Qi/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*) Q2/ In (Z9, +9)…

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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: 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…