Exercise 64. Let G be a group and let H be a subgroup of index n. If G| ≤n!, then G is not simple. Exercise 65. If H is a proper subgroup of An (n ≥ 5), then |G: H| ≥ n (Hint: use the fact that An is simple). Example. The group A5 has no proper subgroup of order greater than 12. In fact by Exercise 65, every subgroup of A5 has index at least 5. The smallest divisor of 60 is 12. Similarly, A6 of order 360 has no subgroup of order greater oro rothor small F

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

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-

A:

Q: Exercise 8.3. (a) If H₁ and H₂ are subgroups of groups G₁ and G₂, respectively, prove that H₁ÐH₂ ≤…

A:

Q: Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are…

A: If the product AB is defined, the entry in row i and column j of AB is the sum of the products of…

Q: Prove the next statement: If G is an abelian group of finite order n, then G has a subgroup of…

A: Answer

Q: Prove the next statement: If H is a subgroup of a finite abelian group G, then G has a group…

A: Introduction: In mathematics, an isomorphism is a type of mapping or function between two…

Q: 4. Let G be a group, and let H be a subgroup of G. Show that if H contains the commutator subgroup…

A:

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

A:

Q: Prove or give counter example Every characteristic subgroup is fully invariant

A:

Q: 42. Let G be a group and C be the collection of all subgroups of G. Is NHec a subgroup of G. What's…

A: Let G be group and C is the collection of all subgroups of G.

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

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

Q: Suppose that a group contains elements of orders 1 through 10.What is the minimum possible order of…

A: Given that, a group contains elements of orders 1 through 10.We have to find the minimum possible…

Q: GH = {g e G|gHg- = H} is a subgroup of G. 12. Let G be a finite group and let primes p and q # p…

A: We have to prove that if G has precisely one proper Sylow P subgroup, it is a normal subgroup so G…

Q: If a group H has finitely many subgroups, does this mean that H is finite group? Justify your answer…

A:

Q: Classify groups of order 2p as best as you can. Give a proof for your assertions and, when…

A:

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

A:

Q: Let G = Z/36Z. Find all subgroups of G, describe all containments between these subgroups, and draw…

A:

Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

A:

Q: 4. Let G be a finite group of order 4 containing no element of order 4. Explain why every…

A: See below

Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4

A: Given: A Sylow 2-subgroup of S4To find: Describe an isomorphism from this group to D4.

Q: Definition 4.24: Let G be a group, and let Ω be the set of subgroups of G. Let G act on Ω by…

A: We prove our claim by using definition of NG(H).

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: In S3, find an example of an H < G, and g1, g2 ∈ G such that the two conjugate subgroups are not…

A: Let H be a subgroup of G Then a H1={ghg | g,h belongs to G,H respectively}. is a conjugate subgroup

Q: How much cyclic groups does have U(15

A: Cyclic group  is a group that is generated by a single element.

Q: Take its Suberou P4 graup.of oder sir and order three, then. ৮ वेराण््डयमड jind the distinct left…

A: Take the group G as ℤ6 ( group of integers under addition modulo 6) ℤ6=0,1, 2, 3, 4, 5 Take H=0, 2,…

Q: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

A:

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

A:

Q: 2. Let G be a group and let H be a subgroup of G. Define N(H) = { x = G | xHx™¹ = H}. Prove that…

A:

Q: Prove that if H is a subgroup of a finite group G of index p, where p is the smallest prime that…

A: This question is about group theory.

Q: Let G and H be two groups, , & € Hom (G, H), and A C G be a nonempty subset. (a) Prove ((A)) = (…

A: We have proved (a) in two steps, showing both the subset relation one by one. For (b) we have used…

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: An environmental engineer wants to evaluate three different methods for disposing of nonhazardous…

A: GIVEN

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…

Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

A:

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

A:

Q: 3. Show that if G is a cyclic group then every subgroup of G is cyclic.

A: Subgroup of a cyclic group