A group G has come to talk to you. You find out that G has 100 = 22 x 52 elements and that G has exactly one subgroup of order 10 whose name is Н. (a) Remind yourself, by rereading Example 10.7, why H must be a nor- mal subgroup of G. (b) Let PE Syl;(G). What can you say about |P|, |H^P|, and |(H, P)|? Prove your assertions. (c) The group G is guaranteed to have subgroups of which sizes?

Q: 16

A: It is known that in the group Zn , if  d divides n , then Zn has a unique subgroup of order d and…

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

A:

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: Let : G H be a homomorphism. Let (G) = {(g) | ge G}. Show that (G) is a subgroup of H.

A:

Q: Is {0, 2, 4, 6} a subgroup of Zg? Justify your answer.

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: (a) (Z4, +). (b) (Z5, .). (c) (Z5, +). (d) (Z4, .). (e) (Z6, .).

A:

Q: 11. Consider the group D, For the subgroup H = {(1), (12)(34)}, write all cosets that can be formed.…

A: D4=Dihedral group of order 4H=1,1234=R0,V and H=2To form the cosets.

Q: Explain why the only simple, cyclic groups are those of prime order.

A: Proof: Let G be a simple group with |G|>1. We want to prove that G is a cyclic group of prime…

Q: Let n E Z be an integer number and let H = (n) = {rn,r E Z} i.e\ H = (n) = {.., -3n, –2n, -n, 0, n,…

A:

Q: Given that G is a group and H is a subgroup. What is the result of (b^-1)^-1 if b is an element of…

A: Given that     G is a group and H is a subgroup of G.    Inverse of an element:  Let G be a group…

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: In (Z,+) determine ({2,4,5}), the subgroup generated by the set {2,4,5). (As always, please provide…

A: In  , we have to determine , the subgroup generated by the set .We know thatDefinition : Given a…

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

A:

Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

A:

Q: Let G = . The smallest subgroup of G containing a^10 and a^12 is generated by * O a^12 a^4 a^2 a^6

A:

Q: The quaternions is the group Q of order 8 consisting of the matrices in GL2(C) Q = {E, A, A², A³, B,…

A:

Q: 2. Which of the following are groups? For those which are not groups, explain why not (it is enough…

A: 2)(i)  under   is  not  a  group  since which has  no  inverse  under    (ii) Let      Then,   is…

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: 4

A: We use theorem of order of group and order of elements of group .

Q: Let G=U(18) and H={1,17} be subgroup of G. The number of distinct left cosets a of H in G is: * 3.

A: Given G=U(18)  H ={1,17} We need to find the number of distinct left cosets of H in G

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: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: The solution is given as