[Number Theory] How do you solve question 3? The second picture is for definitions. Thank you!Additional information:We say that a group G is abelian if its elements commute, that is, gh = hg for all g, h in GUn is an abelian group under multiplication mod (n) DefinitionIf Un is cyclic then any generator g for Un is called a primitive root mod (n).This means that g has order equal to the order (n) of Un, so that the powersof g yield all the elements of Un. For instance, 2 and 3 are primitive rootsmod (5), but there are no primitive roots mod (8) since Us is not cyclic.Finding primitive roots in Un (if they exist) is a non-trivial problem, andthere is no simple solution. One obvious but tedious method is to try each ofthe (n) units a E Un in turn, each time computing powers a mod (n) to findthe order of a in Un; if we find an element a of order (n) then we know thatthis must be a primitive root. The following result is a rather more efficienttest for primitive roots:Lemma 6.4An element a E Un is a primitive root if and only if a(n)/9 ‡ 1 in Un for eachprime q dividing (n). 1. (i) Show by hand that 2 is a primitive root mod 19.(ii) Find² a complete set of primitive roots mod 19.2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not aprimitive root mod p.3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and onlyif ord, (2)| (q-1) and ord, (2)| (p-1). Here, if gcd(a, p) = 1, we use the notation ord, (a)for the order of the element [a], in the group Up.4. (i) Show that 2 is a primitive root mod 11² = 121. Hint: Use the criterion given byLemma 6.4 and a calculator. Conclude that 2 is a primitive root mod 11e for any e.(ii) How many elements of order 5 are there in the group U1331? Note that 1331 = 11³.(iii) Find all elements of order 5 in U1331.¹1.e., do not use a calculatorHere you may use a calculator to compute residues mod 19.

Note: According to guidelines I will answer first question only.

Answered: 1. (i) Show by hand that 2 is a…

Math

Advanced Math

1. (i) Show by hand that 2 is a primitive root mod 19. (ii) Find² a complete set of primitive roots mod 19.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: QUESTION 4 a) Find the order of each element in Z15/(5). b) Find n such the this group is isomorphic…

Q: I need help understanding attached question for abstract algebra. My answer teacher said dosnt…

A: Given two permutations in a symmetric group are

Q: 1. Let G be an abelian group with the identity element e. If H = {x²|x € G} and K = {x € G|x² = e},…

Q: 6

A: To show that there is no isomorphism between the two direct sums , by considering orders of elements…

Q: Prove that a group of order 7is cyclic.

A: Solution:-

Q: Question 1. Let G is a finite group and ø : G → G' be a surjective homomorphism (that is, o is a…

A: To prove: (a)Every element in ℚℤ has finite order. (b) Every non identity element in ℝℚ has infinite…

Q: Question 4. An element g in a group G is called a square if g = a? for some a e G. Suppose that G is…

Q: Exercises 3(a) Prove that the groups $ and D6 are isomophic ; (e) Preve that the additive and the…

Q: Question 2. Let G be a finite group, H 3 G, N 4G, and gcd(|H|, |G/N|) = 1. Prove that H 3N.

Q: Example 3: Let Dg be the set of planar symmetries of a square: Show that (Dg, o) is a finite…

A: For a set G with binary operation '∘' is said to form a group if it satisfies the following…

Q: QUESTION 3 Find the oder of the indicated element in indicated quotient group. a) 2 + (8)in Z12/ (8)…

Q: Let G = Z6 and H = (3). Write out he group table for the quotient group G/H.

A: Let G = Z6 and H = <3>. Write out he group table for the quotient group G/H.

Q: Write the group table for Z₂ XS2.

Q: Question 5. (a) Prove that every element in Q/Z has finite order. (b) Prove that every non-identity…

Q: QUESTION 11 Let G= be a cyclic group of order 8. Find all the distinct subgroups of G and list the…

Q: Question 2. Let G be a finite group, H < G, N 4G, and gcd(|H|,|G/N|) = 1. Prove that H < N.

Q: QUESTION 6 Prove that a cyclic group G= is Abelian.

A: Given cyclic group G = <a>

Q: 5. (a) Give the definition of a group. (b) Show that (Q, +) is an abelian group. (c) Let R= (Z,…

A: Our guidelines we are supposed to answer only first 3 subpart of the question. Kindly repost other…

Q: 4. (i) Show that 2 is a primitive root mod 112 = 121. Hint: Use the criterion given by Lemma 6.4 and…

Q: I denotes the set of real numbers and (*) is an operation on R such that a * B = a +B+ aß for ali a,…

Q: QUESTION 2 Each quotient group is cyclic; to which Z, is it isomorphic? a) Z16/ (5> b) Z18/(12)…

A: Here given that each quotient group is cyclic and we have to find to which zn it is isomorphic.

Q: estion 3. Suppose G is a group with order 99. Prove that G must have an element of order 3.

Q: QUESTION 9 Find up to isomorphism all Abelian groups of order 36.

A: We have to find the all abelian groups of order 36 up to isomorphism.

Q: 18

A: Obtain the two subgroups of order 12 as follows:

Q: Consider the group G = {x € R]x # 1} under the binary operation : *• y = xy – x-y +2 The identity…

Q: Using the Theorem of Lagrange, prove that a group G of order 9 is abelian

Q: Two dice, one green and one yellow, are rolled. Find the conditional probability. (Enter your…

Q: 2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not a primitive…

Q: QUESTION 4 Let G= be a cyclic group of order 20. Find all the elements of order 10 in G.

Q: 4. (i) Show that 2 is a primitive root mod 11² = 121. Hint: Use the criterion given by Lemma 6.4 and…

Q: 3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only if ord,…

A: We shall be using the fact that a^p-1 =1 mod p

Q: 3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only if ord,…

Q: For the differential equations below, what values of ^ provide non-trivial solutions? y" + X²y = 1,…

A: The given differential equation is , y"+λ2y=1 , Y(1)=0, y(2)=1y"+λ2y=1 , Y(0)=y(1)=0

Q: 2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not a primitive…

A: A element a is called a primitive root modulo p if the order of the element a is (p-1)

Q: Prove or Disprove that the Klein 4-group V4 is isomorphic to Z4.

A: The Klein 4 Group is a least non cyclic group. All the none identity element of the Group, which…

Q: Consider the set S of ordered pairs of real numbers together with the operation defined by (a, b)*…

A: As per the company rule, we are supposed to solve one problem from a set of multiple problems.…

Q: R denotes the set of real numbers and (*) is an operation on R such that a * B = a +B+ aß for all a,…

Q: . (i) Show by hand that 2 is a primitive root mod 19. ii) Find2 a complete set of primitive roots…

Question

[Number Theory] How do you solve question 3? The second picture is for definitions. Thank you!

Additional information:

We say that a group G is abelian if its elements commute, that is, gh = hg for all g, h in G
Un is an abelian group under multiplication mod (n)

Definition
If Un is cyclic then any generator g for Un is called a primitive root mod (n).
This means that g has order equal to the order (n) of Un, so that the powers
of g yield all the elements of Un. For instance, 2 and 3 are primitive roots
mod (5), but there are no primitive roots mod (8) since Us is not cyclic.
Finding primitive roots in Un (if they exist) is a non-trivial problem, and
there is no simple solution. One obvious but tedious method is to try each of
the (n) units a E Un in turn, each time computing powers a mod (n) to find
the order of a in Un; if we find an element a of order (n) then we know that
this must be a primitive root. The following result is a rather more efficient
test for primitive roots:
Lemma 6.4
An element a E Un is a primitive root if and only if a(n)/9 ‡ 1 in Un for each
prime q dividing (n).

1. (i) Show by hand that 2 is a primitive root mod 19.
(ii) Find² a complete set of primitive roots mod 19.
2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not a
primitive root mod p.
3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only
if ord, (2)| (q-1) and ord, (2)| (p-1). Here, if gcd(a, p) = 1, we use the notation ord, (a)
for the order of the element [a], in the group Up.
4. (i) Show that 2 is a primitive root mod 11² = 121. Hint: Use the criterion given by
Lemma 6.4 and a calculator. Conclude that 2 is a primitive root mod 11e for any e.
(ii) How many elements of order 5 are there in the group U1331? Note that 1331 = 11³.
(iii) Find all elements of order 5 in U1331.
¹1.e., do not use a calculator
Here you may use a calculator to compute residues mod 19.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Introduction

VIEW

Step 2: Calculation

VIEW

Solution

VIEW

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

[Number Theory] How do you solve question 2? The second picture is for definitions. Thank you! Additional information: We say that a group G is abelian if its elements commute, that is, gh = hg for all g, h in G Un is an abelian group under multiplication mod (n)
[Groups and Symmetries] How do you solve question 3, thanks
Abstract Algebra I
8
[Groups and Symmetries] How do you solve question 5, thanks
[Groups and Symmetries] How do you solve Q3, thanks
Kindly answer correctly, Please show the necessary steps 1c
Reading Question 6.2.1. Let G be a cyclic group of order 20. Which of the following claims is true? Select all that apply. (a) G has 5 elements of order 4. (b) G has 4 elements of order 5. (c) G has 1 element of order 2. (d) G contains 10 generators. (e) If a is a generator of G, then the order of ak is equal to the gcd(20, k).
Abstract Algebra I
Abstract algebra Help please
2
Theirem: let (G, *) be a group then 1- the identity element of G is Unique. 2- the inverse of any element a in G is Unige. 3- (₁²)² = a, & V a E G -1 -1 4- (a*b)² = 6¹² * a²¹, ab € G