Group theory Consider the group GL2(Zp) of invertible 2X2 matrices with entries in the field Zp,...

Group theory

Consider the group GL₂(Z_p) of invertible 2X2 matrices with entries in the field Zp, where p is an odd prime.

Zp is an abelian group under addition, the group of unites of Zp is Zp^x, which is an abelian group under multiplication. We say (Z_p , +, ·) is a field.

1. Show that the subset D₂(Z_p) of diagonal matrices in GL₂(Z_p) is an abelian subgroup of order (p - 1)².
1. For A, B ∈GL₂(Z_p), show that A and B are in the same right D₂(Z_p)-coset if and only if there are non-zero elements λ, µ ∈ Z_p such that A can be obtained from B by multiplying the first column by λ and the second column by µ.
2. For A, B∈GL₂(Z_p), show that A and B are in the same left D₂(Z_p)-coset if and only if there are non-zero elements λ, µ ∈Z_p such that A can be obtained from B by multiplying the first column by λ and the second column by µ.
3. Find a matrix A such that the left and right D₂(Z_p) cosets containing A are different, that is, so that D₂(Z_p)A ≠ AD₂(Z_p).
4. Find a non-identity matrix B such that the left and right D₂(Z_p) cosets containing B are the same, that is, so that D₂(Z_p)A = AD₂(Z_p).

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Which of the following are true? (1) The sum of two invertible matrices is invertible (2)...

Which of the following are true? (1) The sum of two invertible matrices is invertible (2) The determinant of the sum is the sum of the determinants (3) The determinant of the inverse is the reciprocal of the determinant (4) An nxn matrix is invertible if and only if its determinant is zero (5) The product of two invertible matrices is invertible (so long as the product is defined) (6) If A is a diagonal 3x3 matrix [a,0,0;0,b,0;0,0,c] then its...

Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2) +...

Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2) + AB + BA + B^(2). Hint: Write out 2x2 matrices for A and B using variable entries in each location and perform the operations on either side of the equation to determine whether the two sides are equivalent.

Two square matrices A and B are called similar if there exists an invertible matrixV suchthatB=V−1AV....

Two square matrices A and B are called similar if there exists an invertible matrixV suchthatB=V−1AV. (a) Explain in detail how similarity is related to the change of basis for- mula. What does similarity imply about the geometric relationship between B and A?

Let M 2,2 be the set of all 2x2 matrices determine whether the following subspaces. A)...

Let M 2,2 be the set of all 2x2 matrices determine whether the following subspaces. A) the set of all 2x2 diagnol matrices B) the set of all matrices with a12 entry C) the set of all 2x2 triangular matrices

How many distinct 2x2 matrices can we have by using the numbers 1,2,3 and 4. Repeating...

How many distinct 2x2 matrices can we have by using the numbers 1,2,3 and 4. Repeating numbers is not allowed. 2!*2! 4*3*4*3 4! How many multiplications are required to compute AB if A is a 4x7 matrix and B is a 7x5 matrix? 35 28 980 140 4^4 Let A be an nxn matrix, What is det(A) if A has a row of zeros? n-1 1 0 n

Group Theory

Suppose G has precisely one subgroup of order 5, and one subgroup of order 7. What is G?

Determine if the vector(s), polynomial(s), matrices are linearly independent in R^3, P3(R), M 2x2 (R). Show...

Determine if the vector(s), polynomial(s), matrices are linearly independent in R^3, P3(R), M 2x2 (R). Show algebraically how you found your answer. a. < 2, 1, 5 > , < -2, 3, 1 > , < -4, 4, 1 > b. x^3 - 3x^2 + 2x +1, -2x^3 + 9x^2 -3, x^3 + 6x c. | 1 2 | | -3, -1 | | -4 2 | , | 2 1|

Question