Answered: 1.) Let G ={1,−1,i,−i} be the group of…

Math

Advanced Math

1.) Let G ={1,−1,i,−i} be the group of 4 complex numbers under multiplication. (b) Is {1,−1} a subgroup of G and why?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.2: Complex Numbers And Quaternions

Problem 50E

See similar textbooks

Related questions

Q: Show that the Galois group of x 4 + x² + 1 is the same as that of x 6-1 and is of order 2.

A: We have to show that the Galois group of is the same as that of x6 -1 and is of order 2.

Q: Prove C*, the group of nonzero complex numbers under multiplication, has a cyclic subgroup of order…

Q: If z1 and z2 are complex numbers such that |21| = |2|, prove that arg(2122) = arg ((21 + z2)²) .

A: This problem is related with complex analysis. Given that z1 and z2 are two complex numbers such…

Q: Let C* be the group of nonzero complex numbers under multiplica- tion and let H = {a+bi E C* | a² +…

Q: 2, + 22 22-21 cos 2, – cos z2 = 2 sin sin 2 Z1 + Z2 Cos 2 21-22 sin z1 + sin z2 = 2 sin sin 3z = 3…

A: Given: z,z1 and z2 belongs to complex numbers. Let us consider, z1=C and z2=D. 1) To prove: cos…

Q: G={1,-1,i,-i} How can I show that this set forms a group with inductive multiplication in the C…

A: Given: G = {1, -1, i, -i}G forms a group with respect to multiplication if it satisfies the…

Q: Consider the set C4 = {i, -1, -i, 1}, the fourth roots of unity in the complex numbers. (a) Make a…

A: The given set is C4=i,-1,-i,1, the fourth root of unity in the complex numbers. We know that Cayley…

Q: Let z be a non-zero complex number. Then Log(z²) = 2Logz ) None of these O if OSArg(z)ST/2 O if…

Q: show that each of the following collections of numbers forms a group under addition (i) The even…

A: Question (i)To Show this is a group, we need to show the following (as per definition of…

Q: * Let G be the group of all non-zero complex numbers under multipilication and let G be the group of…

Q: Prove that the group of complex numbers under addition is isomorphicto R ⨁ R.

A: To prove: The group of complex numbers under addition is isomorphic to ℝ ⊕ ℝ.

Q: G is the group of nth roots of unity under complex multiplication Zn is the group of integers…

A: Consider the given information.

Q: 1. 4. Let a be a complex number such that lal > 2. Prove that 1. and a 1 generate a free group.

A: 4 Given abe a complex number such that a≥2. We have to prove that 10a1 and 1a01 generate a free…

Q: Determine the behaviour of the set of 4 elements {1,-1,i,-i} under the ordinary multiplication of…

A: Given, The set of 4 elements are 1,-1,i,-i.It is to be shown that this set is group.

Q: 15) Prove that the groyp of complex numbers.under addinan is isomarphic to RR.

Q: 1. Consider the nonzero complex numbers z, = 2i, zz = 8i and z; = 7i. (a) Evaluate (z,z,z,) (2) +2,…

A: Introduction: In most cases, the primary value Arg(z) of a complex number z=x+iy is equal…

Q: Find an isomorphism o from the multiplicative group G of nonzero complex numbers to the…

A: We need to find ϕ for the multiplicative group G of non zero complex numbers to the multiplicative…

Q: iii) How are the complex numbers z1 and zz related if arg(z1) = arg(z2)? Buildup examples for this…

A: Given: z1,z2 are two complex numbers such that argz1=argz2, To find: The relation between z1 and z2,…

Q: Prove C*, the group of nonzero complex numbers under multipliation, has a cyclic subgroup of order n…

A: n th root of unity is the cyclic subgroup of order n.

Q: Find the order of G. Find the order of i and -1.

Q: interpret geometrically ||a||=1 in the transformation z -> az (z complex number)

A: Answer given below.

Q: Show that the mapping Φ(a + bi) = a - bi is an automorphism of the group of complex numbers under…

A: Consider the equation

Q: 4. Consider the complex number set { 1,-1,1,-i, (¹+i) (1-i) (-1+i) (-1-i) √√2 √2 √√2 √√2 ordinary…

Question

1.) Let G ={1,−1,i,−i} be the group of 4 complex numbers under multiplication.

(b) Is {1,−1} a subgroup of G and why?

Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.

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

VIEW

Step 2

VIEW

Step by step

Solved in 2 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

Find all homomorphic images of the quaternion group.
9. Find all homomorphic images of the octic group.
Write 20 as the direct sum of two of its nontrivial subgroups.
Exercises 16. Assume that the nonzero complex numbers form a group with respect to multiplication. If and are real numbers and , the conjugate of the complex number is defined to be . With this notation, let be defined by for all in . Prove that is an automorphism of .
In the field of a complex numbers, show that the mapping that maps each complex number onto its conjugate, (a+bi)=abi, is an isomorphism from to .
45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
In Exercises 114, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold. The set of all complex numbers x that have absolute value 1, with operation addition. Recall that the absolute value of a complex number x written in the form x=a+bi, with a and b real, is given by | x |=| a+bi |=a2+b2
11. Show that defined by is not a homomorphism.
Exercises In Exercises, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition that fails to hold. 9. The set of all complex numbers that have absolute value , with operation multiplication. Recall that the absolute value of a complex number written in the form, with and real, is given by.
Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.
help me fast