Let n be a positive integer that is not divisible by 3 and let G
be a group of order 3n. Prove that if x, y ∈ G are elements of
order 3, then y is conjugate to either x or x^(-1).
Answers
In a group G of order 3n, where n is a positive integer not divisible by 3, if x and y are elements of order 3, it can be proven that y is conjugate to either x or x⁻¹. This result highlights the relationship between elements of order 3 in such groups.
To prove that y is conjugate to either x or x⁻¹, we need to consider the properties of groups of order 3n and elements of order 3.
First, since the order of G is 3n, by Lagrange's theorem, the order of any element in G must divide the order of the group. Therefore, the orders of x and y must be divisors of 3.
Since x has an order of 3, it means that x generates a cyclic subgroup of order 3. Similarly, y also has an order of 3 and generates another cyclic subgroup of order 3 in G.
Now, if x and y are elements of the same cyclic subgroup, then they are conjugate to each other. This is because within a cyclic subgroup, every element is conjugate to any other element. Therefore, if x and y are in the same cyclic subgroup, y is conjugate to x.
On the other hand, if x and y are in different cyclic subgroups, we can consider the cosets formed by these subgroups. Since the order of G is 3n and each subgroup has an order of 3, there are exactly n cosets. This means that each element in one subgroup is conjugate to exactly one element in the other subgroup. Therefore, if x and y are in different cyclic subgroups, y is conjugate to x⁻¹.
Hence, we have shown that in a group G of order 3n, if x and y are elements of order 3, y is conjugate to either x or x⁻¹.
Learn more about: Group theory and Conjugacy
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Related Questions
Is 4 10/5, equivalent to 6?YesNo
Answers
The expression 4 10/5 is the same as 4 + 10/5. It has a whole part (4) and a fraction part (10/5).
To find out if this expression is equivalent to 6, we need to simplify the fraction part:
\(\frac{10}{5}=10\div5=2\)We can simplify the fraction part in a whole number (2), so we have that:
\(4\frac{10}{5}=4+\frac{10}{5}=4+2=6\)So the answer is Yes, 4 10/5 is equivalent to 6.
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Answers
Answer:
160 yards
Step-by-step explanation:
i) Multiply: (3.1x10°) x ( 1.5 x 10) = j) Divide: (3.1x10) / ( 1.5 x 10') = Small angle formula is a very useful approximation for angles smaller than about 0.25 radian (~15°). It allows calculation
Answers
i) The multiplication of (3.1x\(10^0\)) and (1.5x10) results in 4.65x\(10^1\).
j) The division of (3.1x10) by (1.5x\(10^{-1\)) equals 2.07x\(10^1\).
i) To multiply numbers in scientific notation, we multiply the coefficients (3.1 and 1.5) and add the exponents (0 and 1) together. In this case, 3.1 multiplied by 1.5 gives us 4.65. Adding the exponents, \(10^0\) multiplied by \(10^1\) results in \(10^1\). Therefore, the final result is 4.65x\(10^1\).
j) When dividing numbers in scientific notation, we divide the coefficients (3.1 and 1.5) and subtract the exponents (1 and -1) from each other. Dividing 3.1 by 1.5 gives us approximately 2.07. Subtracting the exponents, \(10^1\)divided by \(10^{-1\) is equivalent to \(10^{(1-(-1))}\) which simplifies to 10^2. Hence, the result is 2.07x\(10^1\).
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Find the quotient of -132 and 6.
Answers
Answer:
-22
Step-by-step explanation:
Answer:
-22
Step-by-step explanation:
-132 ÷ 6 --- Divide
Given points A(-1,4) and B(x,7), determine the value(s) of x if AB=5cm
Answers
The value of x is either 3 or -5 based on the distance formula.
What is a co-ordinate system?
In pure mathematics, a coordinate system could be a system that uses one or additional numbers, or coordinates, to uniquely confirm the position of the points or different geometric components on a manifold like euclidean space.
Main body:
according to question
Given points A(-1,4) and B(x,7)
Also AB = 5 cm
Formula of distance = \(\sqrt{(y1-y2)^{2}+(x1 -x2)^{2} }\)
here by using points ,
5 = \(\sqrt{(x+1)^{2} +(7-4)^{2} }\)
taking square on both side ,'
25 = \((x+1)^{2} +3^{2}\)
25-9 = (x+1)²
16 = (x+1)²
taking square root on both sides,
x+1= ±4
x = 4-1 = 3 or x = -4-1 = -5
Hence value of x is either 3 or -5.
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PLEASE HELP MY ASSIGNMENT IS DUE IN 3 HOURS!!!!
Answers
Answer:
\(\sqrt{2}\)
Step-by-step explanation:
since it is a 90-45-45 triangle, both of the legs are equal
You draw 2 diameters through the center of the circle. The 2 diameters must be perpendicular. The vertexes of the square are the 4 points where the diameters meet the circles.
Answers
Answer:
square
(ii) If the diameters are perpendicular to each other, then by joining the ends of two diameters, we get a square.
calculate the taylor polynomials 2 and 3 centered at =0 for the function ()=16tan().
Answers
Taylor polynomials of degree 2 and 3 centered at x = 0 for the function f(x) = 16tan(x) are:
P2(x) = 16x + 8x^2
P3(x) = 16x + 8x^2
To find the Taylor polynomials centered at x = 0 for the function f(x) = 16tan(x), we can use the Taylor series expansion for the tangent function and truncate it to the desired degree.
The Taylor series expansion for tangent function is:
tan(x) = x + (1/3)x^3 + (2/15)x^5 + (17/315)x^7 + ...
Using this expansion, we can find the Taylor polynomials of degree 2 and 3 centered at x = 0:
Degree 2 Taylor polynomial:
P2(x) = f(0) + f'(0)(x - 0) + (1/2!)f''(0)(x - 0)^2
= 16tan(0) + 16sec^2(0)(x - 0) + (1/2!)16sec^2(0)(x - 0)^2
= 0 + 16x + 8x^2
Degree 3 Taylor polynomial:
P3(x) = P2(x) + (1/3!)f'''(0)(x - 0)^3
= 0 + 16x + 8x^2 + (1/3!)(48sec^2(0)tan(0))(x - 0)^3
= 16x + 8x^2
Therefore, the Taylor polynomials of degree 2 and 3 centered at x = 0 for the function f(x) = 16tan(x) are:
P2(x) = 16x + 8x^2
P3(x) = 16x + 8x^2
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find the area of the entire surface of the prism.
5 4 3
Answers
Answer:
A=94
Step-by-step explanation:
Length: 5
Width: 4
Height: 3
Shape: Rectangle prism
Solved for surface area
Formula: A=2(wl+hl+hw)
Formula: L*W*H
Answer: A=94
Hope this helps.
Combining like terms of 10x - 3+ 11 - 2x + 3x, simplifies the expression to
Answers
Answer:
-9x+8
Step-by-step explanation:
15x - 3+11 should be the answer
Given two points which represent the endpoints of the diameter of a circle, which of the following statements is true?
A.
The x-coordinate of the center of the circle must be the same as at least one of the x-coordinates of the given endpoints of the diameter.
B.
The center of the circle is the midpoint of the given endpoints of the diameter.
C.
The center of the circle cannot be found without additional information.
D.
The y-coordinate of the center of the circle must be the same as at least one of the y-coordinates of the given endpoints of the diameter.
Answers
The correct statement is B. The center of the circle is the midpoint of the given endpoints of the diameter.
In a circle, the center is located at the midpoint of any diameter. A diameter is a line segment that passes through the center of the circle and has its endpoints on the circle. Therefore, if we are given the endpoints of a diameter, we can determine the center of the circle by finding the midpoint of these endpoints. This means that the x-coordinate of the center will be the average of the x-coordinates of the endpoints, and the y-coordinate of the center will be the average of the y-coordinates of the endpoints.
Option A is not necessarily true because the x-coordinate of the center may or may not be the same as the x-coordinates of the given endpoints.
Option C is incorrect because the center of the circle can be found by determining the midpoint of the diameter.
Option D is not necessarily true because the y-coordinate of the center may or may not be the same as the y-coordinates of the given endpoints.
Therefore, the correct statement is B. The center of the circle is the midpoint of the given endpoints of the diameter.
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I need answer for this problem
Answers
Answer:
the answer is B
in radians it is 10 pi/3
Find the 7th term of the geometric sequence whose common ratio is 1/2 and whose first term is 2.
Answers
Answer:
0.03125
Step-by-step explanation:
\(a_{n}\) = \(a_{1}\)\((r)^{n-1}\)
\(a_{7}\) = (2)\(\frac{1}{2} ^{(7-1)}\)
\(a_{7}\) = (2)\(\frac{1}{2} ^{6}\)
\(a_{7}\) = 0.03125
Helping in the name of Jesus.
in an academic contest, correct answers earn 12 points and incorrect answers lose 5 points. In the final round, School A starts with 165 points and given the same number of correct and incorrect answers. School B starts with 65 points and given no incorrect answers and the same number of correct answers as School A. The game ends with the Schools tied.
Part A: Which equation models the scoring in the final round and the outcome of the contest?
A.) 12x + 5x - 165 = -12x + 65
B.) 12x - 5x + 165 = 12x + 65
C.) 5x - 12x + 165 = 12x + 65
D.) 12x - 5x - 165 = 12x +65
Part B: How many answers did each school get correct in the final round?
Answers
Answer:
part a is b part b is 230 tell me if im wrong
Step-by-step explanation:
Step-by-step explanation:
Let x represent the number of correct answers School A gave.
Then x is also the number of incorrect answers School A gave and the number of correct answers School B gave.
Final score of School A
= 165 + 12x - 5x.
Final score of School B
= 65 + 12x
Hence the correct equation is
12x - 5x + 165 = 12x + 65. (B)
=> 7x + 165 = 12x + 65
=> 5x = 100
=> x = 20
Hence Schools A and B got 20 correct answers each.
Rewrite the problem using substitution and simplify the expression using order of operations and show your work.
3. Evaluate:
Suppose you want to buy something for $60, and you have $15 saved up so far. Then your grandmother calls and says she will chip in for your purchase. She doesn’t tell you the amount of money she will give you yet, so you just consider it x dollars.
Answers
The algebraic expression for the scenario is 15 + x = 60
How to evaluate the algebraic expression for the statement?From the question, we have the following parameters that can be used in our computation:
Amount saved up = $15
Cost of item = $60
Amount gifted by your grandma = x
The above parameters means that the mathematical statement is an equation statement
So, we have the following representation
Amount saved up + Amount gifted by your grandma = Cost of item
Substitute the known values in the above equation, so, we have the following representation
15 + x = 60
Hence, the algebraic expression is 15 + x = 60
The question does not require that the equation be solved
However, when it is solved, we have:
15 + x = 60
Subtract 15 from both sides
x = 45
So, the amount is 45
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Area involving
A rectangular paperboard measuring 35 in long and 24 in wide has a semicircle cut out of it, as shown below.
Find the area of the paperboard that remains. Use the value 3.14 for x, and do not round your answer. Be sure to include the
correct unit in your answer.
24 in
35 in
0
808
in
X
in² in³
Answers
The area of the paperboard that remains is 613.92 square inches.
To find the area of the paperboard that remains after a semicircle is cut out, we need to calculate the area of the rectangular paperboard and subtract the area of the semicircle.
The rectangular paperboard has dimensions of 35 inches long and 24 inches wide. Therefore, the area of the rectangular paperboard is:
Area_rectangular = length * width = 35 in * 24 in = 840 in²
Now, let's calculate the area of the semicircle. The semicircle is cut out of the rectangular paperboard, and the diameter of the semicircle is equal to the width of the rectangular paperboard (24 inches).
The formula to calculate the area of a semicircle is:
Area semicircle = (π * r²) / 2
where r is the radius of the semicircle.
Since the diameter of the semicircle is 24 inches, the radius is half of that, which is 12 inches.
Plugging in the values into the formula, we get:
Area_semicircle = (3.14 * 12²) / 2 = (3.14 * 144) / 2 = 226.08 in²
Finally, to find the area of the paperboard that remains, we subtract the area of the semicircle from the area of the rectangular paperboard:
Area remaining = Area rectangular - Area semicircle = 840 in² - 226.08 in² = 613.92 in²
Therefore, the area of the paperboard that remains is 613.92 square inches.
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if a cup has a diameter of 8 centimeters and a height of 12 centimeters , how much juice will the cup hold.
Answers
The amount of juice the cup can hold given that the cup has diameter of 8 centimeters and a height of 12 centimeters is 602.88 cm³
How do i know the amount of juice the cup can hold?To know the amount of juice the cup can hold, we shall obtain the volume of the cup.
We shall use the formula for obtaining volume of cylinder to obtain the volume of the cup. Details below:
Diameter of cup = 8 cmRadius of cup (r) = diameter / 2 = 8 / 2 = 4 cmHeight of cup (h) = 12 cmVolume of cup (V) =?Volume = πr²h
Volume = 3.14 × 4² × 12
Volume = 3.14 × 16 × 12
Volume = 602.88 cm³
Thus, we can conclude from the above calculation that the amount of juice the cup can hold is 602.88 cm³
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Sophia bought a bag of 20 cough drops for
$3.59. What is the cost per cough drop?
Help I am really confused
Answers
Divide total price by quantity:
3.59/20 = 0.1795
The price for one is $0.1795
If the original quantity is 8 and the new quantity
is 2, what is the percent decrease?
Answers
If the original quantity is 8 and the new quantity is 2, then the correct answer is 75%.
How did we figure this out?
For this question we need to subtract and multiply the numbers. We know that 2 = 25% of 8 so:
\(\boxed{8-2=6}\\\boxed{6/2=3}\)
We are going to take that 25% and multiply it with 3 to get are final answer.
What is the missing number of 25 and 3?\(\boxed{25*3=75}\\\boxed{So,2=75}\)
Therefore, If the original quantity is 8 and the new quantity is 2, then the correct answer is 75%.
a jar contains 11 green marbles, 6 red marbles, and 9 blue marbles. a marble is selected at random, not replaced, and then a second marble is selected. what is the probability of selecting a red marble followed by a green marble?
Answers
Answer:
33/ 338
Step-by-step explanation:
Red = 6
Blue = 9
P.Red = 6/26 = 3/13
P.Green = 11/26
P. Both in turns = 3/13 * 11/26
= 33/338
hope this helps...
Answer:
0.09
Step-by-step explanation:
PLEASE HELP ME THANKS IF U DO (gets reported if link)
Answers
Answer:
5
Step-by-step explanation:
49/9 = 5
PLEASE HELP WILL GIVE BRAINLIEST
Answers
Answer:
no solution is the answer
Step-by-step explanation:
2.8 -3x =12.1 −6x what's x
Answers
PLEASE HELP MY BIRTHDAY WILL BE CANCELLED IF I DONT GET A GOOD GRADE ON THIS
Answers
Answer:
Option 4
Step-by-step explanation:
If the circle is filled in them it will be either ≤, ≥, ≈ .
One circle is open one is closed
A -10 nC charge is located at (x, y) = (1.2 cm , 0 cm).
What is the x-component of the electric field at the position (x, y) = (−4.1cm, 0 cm)?Express your answer to two significant figures and include the appropriate units.
Answers
We can use Coulomb's law to calculate the magnitude of the electric field at a distance r away from a point charge Q:
E = k * Q / r^2
where k is Coulomb's constant, Q is the charge of the point charge, and r is the distance from the point charge.
In this problem, we have a point charge Q of -10 nC located at (1.2 cm, 0 cm), and we want to find the x-component of the electric field at a distance r = 5.3 cm away at position (-4.1 cm, 0 cm).
To find the x-component of the electric field, we need to use the cosine of the angle between the electric field vector and the x-axis, which is cos(180°) = -1.
So, the x-component of the electric field at position (-4.1 cm, 0 cm) is:
E_x = - E * cos(180°) = - (k * Q / r^2) * (-1)
where k = 9 x 10^9 N*m^2/C^2 is Coulomb's constant.
Substituting the given values, we get:
E_x = - (9 x 10^9 N*m^2/C^2) * (-10 x 10^-9 C) / (0.053 m)^2
E_x ≈ -30,566.04 N/C
Rounding this to two significant figures and including the appropriate units, we get:
The x-component of the electric field is about -3.1 x 10^4 N/C (to the left).
find nonzero scalars and so that 4 (2−5) (4 4)=0 for every pair of vectors and . (express numbers in exact form. use symbolic notation and fractions where needed.)
Answers
The scalars 0 and any scalar with magnitude 0 satisfy the equation 4(2−5)(4,4) = 0 for every pair of vectors (4,4).
Vectors are mathematical objects that have both magnitude and direction. They can be multiplied by scalars, which are non-zero real numbers.
In this case, we are looking for non-zero scalars that satisfy the equation 4(2−5)(4,4)=0 for every pair of vectors (4,4).
To begin, we simplify the equation.
The expression 2−5 equals to -3, so
=> 4(-3)(4,4) = 4(-12,-12).
Next, we multiply 4 by -12 to get -48. The expression
=> 4(-12,-12) = (-48,-48).
Now, we need to find non-zero scalars that multiply (-48,-48) to give us a zero vector, which is a vector with magnitude of 0 and direction undefined.
Any scalar that satisfies this condition is a solution. For example, the scalar 0 satisfies this condition because
=> 0 * (-48,-48) = (0,0).
Additionally, any scalar with magnitude 0 will also satisfy this condition because the product of a vector and a scalar with magnitude 0 is always the zero vector.
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∠A and \angle B∠B are complementary angles. If m\angle A=(7x+23)^{\circ}∠A=(7x+23)
∘
and m\angle B=(5x+19)^{\circ}∠B=(5x+19)
∘
, then find the measure of \angle B∠B.
Answers
Since there are complementary angles, the value of angle B is 39°.
Angle A = 7x + 23°
Angle B = 5x + 19°
It should be noted that the total angles in a complementary angle are equal to 90°.
Therefore, we need to add angle A and B together and then equate them to 90°. This will be:
7x + 23° + 5x + 19° = 90°
Collect like terms
12x + 42° = 90°
12x = 90° - 42°
12x = 48°
x = 48°/12
x = 4°
Angle A = 7x + 23° = 7(4) + 23° = 28° + 23° = 51°
Angle B = 5x + 19° = 5(4) + 19° = 20° + 19° = 39°
The value of angle B is 39°.
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x- 0.1x + 2.70 = 56.70 solve for the answer
Answers
Answer:
x=60
Step-by-step explanation:
How many times larger is (1.088 x 10^1) than (8 x 10^-1)
HELP
Answers
The number 1.088 x 10¹ is 13.6 times larger than 8 x 10⁻¹
How many times larger is (1.088 x 10¹) than (8 x 10⁻¹)?To find how many times larger is (1.088 x 10¹) than (8 x 10⁻¹), we just need to take the quotient between these two numbers. To do so remember that when we take the quotient between two powerswith the same base, we just need to subtract the exponents.
Then here we will get:
\(\frac{1.088*10^1}{8*10^{-1}} = \frac{1.088}{8} *10^{1 - (-1)} = 0.136*10^2\)
We can rewrite that as:
1.36*10 = 13.6
Then the first number is 13.6 times larger than the second one.
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On one play, a football team gained 9 yards. On the next play, the football team received a penalty
of -10 yards.
Answers
Answer:
If your asking what the sum of the yards is it would be -1
Step-by-step explanation:
First I would make a number line from -10 to positive 10. Then from zero add nine. Then subtract 10 from 9. Since 10 is bigger than nine it would be a negative number. Hope this helped.