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

Q: (d) Show that Theorem 1 does not hold for n 1 and n = 2. That is, show that the multiplicative…

A:

Q: Suppose that H is a subgroup of Z under addition and that H contains 250 and 350.  What are the…

A: To determine the possible subgroups H satisfying the given conditions

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

A:

Q: 7. Show that 4 is a subgroup of S,

A:

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

A:

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: Prove that there is no simple group of order 300 = 22 . 3 . 52.

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: (H,*) is called a of (G,*) if (H,*) is a group.

A:

Q: 4.14. Show that an element of the factor group R/Z has finite order if and only if it is in Q/Z.

A: Any rational number can be written in the form p/q where p and q are relatively prime integers.Since…

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

A:

Q: Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176, 189, 195, 200 is not simple.

A: To show that the group of order 45 is not simple.

Q: Show that if a group has an even number of elements then there is a element A other than unity such…

A: We have to prove that if a group has an even number of elements then there is aelement A other than…

Q: In a group G,let a,b and ab have order 2.show that ab=ba

A:

Q: Every group of order 4 is cyclic. True or False then why

A: Solution

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

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

Q: What are the three things we need to show to prove that an ordered pair is a group?

A: We have to give the properties of an ordered pair to prove that it is a group.

Q: Prove that there is no simple group of order 525 = 3 . 52 . 7.

A: The prime factors of 525 are 3, 5 and 7. So there are proper normal subgroups of order either 3,5 or…

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

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

Q: 27. If g and h have orders 15 and 16 respectively in a group G, what is the order of (9) n (h)?

A:

Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

A:

Q: Every commutative group has at least element ??

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

Q: 8. Use Caley's table to prove that the set of all permutations on the set X = {1,2,3} is indeed a…

A: A set R together with the binary operations addition is said to be group if it satisfies the given…

Q: The number of generators of a cyclic group of order 213 is * 48 24 144 140

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: 3. List all elements of the cyclic subgroup of Z12 generated by 5

A: Solving

Q: Prove that there is no simple group of order 210 = 2 . 3 . 5 . 7.

A:

Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: Prove that there is no simple group of order 528 = 24 . 3 . 11.

A:

Q: 16. A class consist of 15 men and 20 women, in how many ways can a group of 5 be formed if it has to…

A:

Q: Suppose that <a>, <b> and <c> are cyclic groups of orders 6, 8, and 20,…

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: 8. Give an example of a group G where the set of all elements that are their own inverses does NOT…

A: Let, G,. is a group. Let, G={1,7,17,2,12,3,13} Let, H be a subgroup of G where H={1,7,17,2,12}

Q: If a is an element of order 8 of a group G, and 4 = ,then one of the following is a possible value…

A:

Q: Every finite group is cyclic

A:

Q: Consider the discrete group G of order 8 that has the following Cayley diagram e If we have the…

A: The sequence of operations is fcagec. Each element g of G is assigned a vertex: the vertex set…

Q: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.

A:

Q: Any group of order 520 is simple

A: Simple group:     "A simple group is a nontrivial group whose only normal subgroups are the trivial…

Q: Prove that any group of order 75 can have at most one subgroup of order 25.

A:

Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…

A:

Q: Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the…

A: We know that, Order of the given group is divisible by natural numbers 5,7,8 and 9. So the least…

Q: 6.7 Construct a nonabelian group of order 16, and one of order 24.

A:

Q: 7. Prove that if G is a group of order 1045 and H€ Syl₁9 (G), K € Syl (G), then KG and HC Z(G).

A: 7) Let G be a group of order 1045 and H∈Syl19(G) , K∈Syl11(G).  To show: K⊲G and H⊆Z(G). As per…

Q: (5) Make a list of all group homomorphisms from Z4 → Z8.

A:

Q: (b) Complete the following character table of a group of order 12: 1 3 4 X1 X2 X3 4,

A: The character table of a group of order 12:

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.

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.