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: How many cyclic subgroups does U(15) have?

A: To find Number of  cyclic subgroups does U(15) have

Q: true O false O * -2 2) Let |G| = n and H a subgroup of G then la, + H| = la, + H| = laz * H| = ... =…

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

A:

Q: We consider the group Ss. This is a group with 8! elements. Give an example of a 7-Sylow subgroup.…

A: Consider the group S8. This group contains 8! elements. We know that, if G is a group and p is a…

Q: (P) B (iv) с All D (iii), (iv) (i), (v) (i), (iv), (v) Identity which are subgroups of $10. (i): The…

A: The given permutation  group is  S10 . The permutation group S10 is a set of all permutation  of the…

Q: Find all subgroups of U(7)

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: Every cyclic group is a non-abelian group.

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

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

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: 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: 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: 4

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

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: 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: 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: 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: 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

Q: 2.1.12. Corollary Let (H)be a nomal subgroup of the group (G). There is one -one correspondence…

A: