5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, Cg(A) = A and NG(A) = G. (a) G = S3 and A = {1, (1 2 3 ), (1 32)}.

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: In (Z12, +12) , H, = {0,3,6,9} and H2 = {0,3} are tow subgroups of the group (Z12, +12), but (H, U…

A: We have given   H1=0,3,6,9 and H2=0,3 are two subgroups of the group Z12, +12. We can see here…

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

A:

Q: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

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

A:

Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

A:

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

A:

Q: Explain why a group of order 4m where m is odd must have a subgroupisomorphic to Z4 or Z2 ⊕ Z2 but…

A:

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: 11. Find the cyclic subgroup of D4 generated by µp². What is the order of this subgroup?

A:

Q: (c) Prove that the intersection of any three subgroups is a subgroup while the union of two…

A:

Q: 1- The index 2- infinite group 3- permute elements B prove that (cent G,*) is normal subgroup of the…

A:

Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

A:

Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

A:

Q: (3) Show that 2Z is isomorphic to Z. Conclude that a group can be isomorphic to one of its proper…

A: (2ℤ , +) is  isomorphic to (ℤ , +) .              Define f :(ℤ , +) →(2ℤ , +) by…

Q: 17. Show that every group of order (35)° has a normal subgroup of order 125.

A:

Q: 5. Suppose G is a group of order 8. Prove that G must have a subgroup of order 2.

A:

Q: Show that ( Z,,+,) is a cyclic group generated by 3.

A:

Q: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

A: Given question:   Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: The Sylow theorems are significant in the categorization of finite simple groups and are a key…

Q: Compute the center of generalized linear group for n=4

A: To find - Compute the center of generalized linear group for n=4

Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

A: We need to prove that H=1+2n1+2m∋n,m∈Z is a subgroup of Q-0 A subset W of a group V is said to be a…

Q: 2. Use one of the Subgroups Tests from Chapter 3 to prove that when G is an Abelian group and when n…

A:

Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).

A:

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

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: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

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

A:

Q: Suppose x is an element of a cyclic group of order 15 and x3 = x7 = x°. Determine |x13].

A: According to a theorem in group theory , If G is a finite group and a∈G be an element in the group…

Q: 4

A: To identify the required cyclic subgroups in the given groups

Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

A:

Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…

A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…

Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: The group (Z6,6) contains only 4 subgroups

A:

Q: Show that 40Z {40x | * € Z} is a subgroup of the group Z of integers. Note: Z is a group under the…

A:

Q: (a) Compute the list of subgroups of the group Z/45Z and draw the lattice of subgroups. (prove that…

A: In the given question we have to write all the subgroup of the group ℤ45ℤ and also draw the the…

Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…

Q: The group (Z6,+6) contains only 4 subgroups

A:

Q: If H is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…

A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.

Q: (5) Show that in a group G of odd order, the equation x² = a has a unique solution for all a e G.

A:

Q: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

Q: What could the order of the subgroup of the group of order G| = 554407

A: We find the possible order of all the subgroups of the group G, where |G|=55440 by using Lagrange's…

Q: If H₁ and H₂ be two subgroups of group (G,*), and if H₂ is normal in (G,*) then H₂H₂ is normal in…

A: When a non-empty subset of a group follows all the group axioms under the same binary operation, the…

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: 7. You have previously proved that the intersection of two subgroups of a group G is always a sub-…

A:

Q: 2. If H and K are subgroups of an abelian group G, then HK = {hk | h e H and k e K} is a subgroup of…

A:

Q: 1. Let G be a group and let H, H, .. H, be the subgroups of G. The ...

A:

Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

Linear Correlation

A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related. 

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.