Answered: 13. For each of the following values of…

13. For each of the following values of n, find all subgroups of the group G described in Exercise 11, and state their order. а. п 3D 7 b. n = 5 с. п 3D 11 d. n = 13 е. п %3D 17 f. n = 19

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 13E: Exercises 13. For each of the following values of, find all subgroups of the group described in...

See similar textbooks

Related questions

Q: In doing plannned comparisons, we use a. The T test for independent groups with MS within in the…

A: Introduction: When there is significant difference between the means of several treatments, the…

Q: 7. Draw the subgroup diagrams of the following groups: (a) Z180

A: Since you have asked multiple questions, we will solve first one for you. If you want other question…

Q: Find the order of the following elements in the given groups: a. The order of sr15 in D50 is o…

Q: compute the 3 -sylow subgroups of S5

A: n3 (S5 ) =(1+3k) /40

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

A: In given Cayley Table : * 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1…

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

A: According to our company's guidelines I can only answer first question since you have asked multiple…

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

Q: Which of the following is the identity element in the group Z/14Z × µs O (147,0) O (142,1) O…

Q: B. Write at least five elements of each of the following cyclic groups. 1. 25Z under multiplication…

A: 25Z under multiplication: {-50, -25, 25, 50, 75}

Q: 5. Find the number of generators of the cyclic group Z15

A: To find the number of generators of the cyclic group ℤ15.

Q: Give the subgroup diagram for each of the groups: (a) Z24 (b) Z36-

Q: 2. Determine the number of elements of order 15 in the group Z75 ZL20- Also determine the number of…

Q: 12. Find the index of (3) in the group Z24.

A: This problem is based on abstract algebra.

Q: Determine the number of elements of order 15 and the number ofcyclic subgroups of order 15 in Z30 ⨁…

Q: Let p1, p2 ......pn be distinct primes. Up to isomorphism, how manyAbelian groups are there of order…

A: Given Data The distinct prime is p1, p2…………. pn. The number of Abelian groups of the order pm, so…

Q: 6. Let a be an element of order n in a group and let k be a positive la| = integer. Then 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: 1. Determine all subgroups of the group (U13, ·)

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

Q: The elements of order 8 in the group (Zg ,+) is a)4 b) 5 c)6 d) 8

A: In this question we have to find the number of elements of order 8. (a) 4 is correct option

Q: 15. Suppose that N and M are two normal subgroups of a group G and that NO M = {e}. Show that for…

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

A: 1

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

Q: How many subgroups of the group G = (Z30, +30)? O a. 4 O b. 7 O c. 5 O d. 6 O e. 8

A: We know that τ(n) is number of positive divisors of n Also τpaqbrc = (a+1)(b+1)(c+1) , where p , q ,…

Q: 7. (a) What is the order of (1, (123)) in Z4 × S3 ? (b) Write down all the distinct cosets of the…

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

Q: How many subgroups of the group G = (Z30, +30)? а. 8 O b. 6 С. 5 O d. 7 O e. 4

Q: Find four different subgroups of S4 that are isomorphic to S3.

A: If there exists an isomorphism between the subgroups they are called isomorphic ( Isomorphism is…

Q: 4-Let (Z12, +12) be a group and let S={4,6}, find subgroup H generated by S. if exist

A: I have used the definition of subgroup generated by a subset.

Q: 35. Determine the subgroup lattice for Z, where p is a prime and n is some positive integer.

A: To determine the subgroup lattice for Zpn Where p is a prime and n is some positive integer. A…

Q: 4

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

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: Match each entry in the first column with the corresponding entry in the second column. (Multiple…

A: We use the results of cyclic group to give the answers.

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

Q: 2. What is the order of the element 32 in the group Z36?

A: Modular groups are cyclic groups. A group G is cyclic if G=<g> for some g in G, where…

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: 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: Which of the following is nontrivial proper sub- group of Z4? {0, 2} O Diophantus of Alexandria {0,…

Q: In proving that G/N is a group where do we first use the fact that N Is ha Select one: a. The invers…

Q: The group ((123)) is normal in the symmetry group S3 and alternating group A4.

Q: What is the relationship between a Sylow 2-subgroup of S4 and the symmetry group of the square? that…

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

A: From given table it is clear that

Q: Let (Z's. ) be the multiplicative group modulo 54. a. Is this group cyclic? How many generators does…

A: (a) Zn is a cyclic group of order n. Here n=54. So, Z54 is a cyclic group. The number of generators…

Q: Consider S4 and its subgroups H = {i,(12)(34),(13)(24),(14)(23)} and K = {i,(123),(132)}. For a =…

A: Note: We are using the simple procedure that is by direct calculation. We are given the group S4 and…

Q: Compute the orders of the following groups.a. U(3), U(4), U(12)b. U(5), U(7), U(35)c. U(4), U(5),…

Q: (b) How many elements of the permutation group Se map 2 to 2 and 5 to 5, while the re- maining…

Q: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

Q: (b) Consider the incomplete character table for a group given below:

A: The table for the incomplete character is, 1 a b c d x1 1 1 1 1 1 x2 1 1 -1 -1 1 x3 1 1…

Q: Give the Cayley table for the group Z2 under multiplication modulo 12.

A: Since , We know that Z12 = 0,1,2,3,4,5,6,7,8,9,10,11 and…

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:

Question

Group G is the nonzero elements of group Z_nwith respect to multiplication

13. For each of the following values of n, find all subgroups of the group G described in
Exercise 11, and state their order.
с. п %3D 11
f. n = 19
a. n = 7
b. n = 5
d. n = 13
е. п 3D 17

Expert Solution

Step 1

Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If you want remaining sub-parts to be solved, then please resubmit the whole question and specify those sub-parts you want us to solve.

(a) consider the provided question,

$n = 7$

$z_{7} = \{0, 1, 2, 3, 4, 5, 6\} H e r e, U (7) = \{0, 1, 2, 3, 4, 5, 6\} a s t h e n o n z e r o e l e m e n t o f z_{7} w i t h r e s p e c t t o m u l t i p l i c a t i o n,$

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.
The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
what are the group means for each of the three groups？
What is the order of the group ℤ10 x ℤ15 x U(10) /〈(9, 10, 7)〉?
Compute the orders of the following groups.a. U(3), U(4), U(12)b. U(5), U(7), U(35)c. U(4), U(5), U(20)d. U(3), U(5), U(15)On the basis of your answers, make a conjecture about the relationshipamong |U(r)|, |U(s)|, and |U(rs)|.
Create steps along with justifications to verify that in a group system (G, +) the following property holds:For any two elements from G, ‘a’ and ‘b’, -(a + b) = (-b) + (-a). In other words, the claim is that (-b) + (-a) plays the role of the inverse of a + b. Create steps to show that the element (-b) + (-a) does, in fact, play the role of an inverse to the element a + b, i.e., show that:i. (a + b) + ( (-b) + (-a) ) = e, where e represents the identity in the group; andii. ( (-b) + (-a) ) + (a + b) = e.
If a group has exactly 24 elements of order 6, how many cyclic subgroupsof order 6 does it have?
explain in detail if the following statement is true or false: " In the permutation group S5, the product of 2020 4-cycles is an even permutation."
How many nonisomorphic frieze groups are there?
In doing plannned comparisons, we use a. The T test for independent groups with MS within in the denominator and DF equal to those of MS within b. The t-test for independent groups with MS within in the denominatorc. The T test for independent groups, unaltered D. df equal to those of MS within
Show that the elements of S9 which send the numbers 2, 5, 7 among themselves form a subgroup of S9. what is the order of this subgroup?