If G is an infinite group, what can you say about the number ofelements of order 8 in the group? Generalize.

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

A:

Q: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345

A: For group, 2*3*4=(2*3)*4.

Q: 28. Is every group a cyclic? Why?

A:

Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: True or False. Every group of order 159 is cyclic.

A: According to the application of the Sylow theorems, it can be stated that: The group, G is not…

Q: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

A:

Q: 11. Prove that every Cayley table is a Latin square for a group. That is, each element of the group…

A: To prove, each element of the group appears exactly once in each row and each column of a Cayley…

Q: How many elements of a cyclic group with order 10 have order dividing 5?

A:

Q: A group that also satisfies the commutative property is called a(n). (or abelian) group. A group…

A: A group that also satisfies the commutative property is called a(n) commutative group (or abelian)…

Q: Show that if n > 6, then the symmetric group Sn contains an element of order greater than n

A: According to the problem, we have

Q: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

Q: 25. o: Z4 → Z12

A: Homomorphism : Let us consider a map f: V→W then f is said to be homomorphism if for all v,u∈V…

Q: If a is an element of order 8 of a group G, and

A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e          where e is an…

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: Suppose that H is a subgroup of Z under addition and that H contains250 and 350. What are the…

A:

Q: The following is a Cayley table for a group G. The order of4 is: 2 1 3 3 4 1 4 1 4 1 5. 2 512 m 45…

A: By observing the Cayley table :

Q: many

A: We have to find the number of generator of the given group of order

Q: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

A:

Q: Is the set of numbers described below a group under the given operation? Whole numbers; addition ...

A: Introduction: A group is defined as a set that has a binary operation that allows any two elements…

Q: Suppose that H is a subgroup of a group G and |H| = 10. If abelongs to G and a6 belongs to H, what…

A: Given: H is a subgroup of a group G and |H| = 10 To find: If a belongs to G and a6 belongs to H,…

Q: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.

A:

Q: Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity…

A: Calculation:Obtain the Cayley table for the set S = {5,15,25,35} under the multiplication modulo 40…

Q: How many subgroups of a not abelian group of order 6 is non-cyclic? Select one:

A: Given: The order of the group = 6.

Q: How many ways can a group of five be chosen to work on a project? As in Example 9.5 4, since the of…

A: Given that, the computer programming team has 9 members.

Q: 1. Determine all subgroups of the group (U13, ·)

A: The sub group of U13 is to be determined.

Q: 1. Show that every group of prime order is simple.

A:

Q: How many elements of a cyclic group with order 14 have order 7?

A:

Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

A:

Q: Every commutative group has at least element ??

A: Every commutative group has at least               element ?  We know that , every commutative group…

Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is a…

Q: 50. How many proper subgroups are there in a cyclic group of order 12? A 4 в з с 2

A: see 2nd step

Q: Let α,β ESs ( a = (1,8,5,7)(2,4) and B= (1,3,2,5,8,4,7,6). Compute aß. Symmetric group) where

A:

Q: Show that there are no simple groups of order 255 = (3)(5)(17).

A:

Q: Let p be a prime number and (G, *) a finite group IGI= p?. How can you prove that the group (G, *)…

A:

Q: Suppose that G is a cyclic group and that 6 divides |G|. How manyelements of order 6 does G have? If…

A:

Q: Suppose H and K are subgroups of a group G.  If |H|=12 and |K| = 35, find |H intersected with K|.…

A:

Q: 1. There is no simple group of order 200.

A: Solution:-As per guidelines I submit first question only

Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

Q: What is the smallest positive integer n such that there are two nonisomorphicgroups of order n? Name…

A: Non-isomorphic groups: Groups that have different Sylow-2 groups are non-isomorphic groups.

Q: Suppose G is a cyclic group with an element with infinite order. How many, elements of G have finite…

A:

Q: Every finite group is cyclic

A:

Q: What limits will include the middle 80% of the group?

A: The required calculation can be defined by the formula Z=x-μσP(a≤Z≤b)=P(b≤Z)-P(a≤Z) Given, What…

Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…

A: The identity element in a subgroup H of a group G must be the same as the identity element in G.

Q: Find an example of a noncyclic group, all of whose proper subgroupsare cyclic.

A:

Q: Is Z6 (set of integers modulo 6) a group under addition?  If your answer is yes, prove your answer.…

A:

Q: Consider the multiplicative group ℤ∗8663. a) How many primitive elements does this group have? b)…

A: (a): we have 8540 primitive elements in this group (b): probability is:0.98

Q: Prove that there is no simple group of order 315 = 32 . 5 . 7.

A: Prove that there is no simple group of order 315=32·5·7.

Q: a) How many subgroups does (Z10,O) have? What are they? b) How many subgroups does (Z74,Ð) have?…

A: 2. a) Consider the group ℤ10, ⊕. The elements of the above group are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…