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).

Expert Solution

Colby Messinger answered 2 years ago

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

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