Find the elementary divisors and invariant factors of .
To find the elementary divisors and invariant factors of the group \( \mathbb{Z}_8 \times \mathbb{Z}_{12} \times \mathbb{Z}_{15} \), we first need to factorize each component of the direct product into its primary decomposition.
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Compute the following:
a).
b).
Show that, to prove that SP2(R) acts transitively on GL2(R), it is enough to show that, for any vector , there is a
symplectic matrix with
as the first column. (Hint: For any matrix A, what is
?)
Suppose that is
matrix where A,B,C and D are
matrices. Show that M is symplectic if and only if the following conditions are satisfied:
(Hint: Use block matrix multiplication.) Also, check that the matrix , where A is a
orthogonal matrix, is a symplectic matrix.
Find the elementary divisors and invariant factors of .
Let and let
be algebraic over F of degree m and n, respectively. Show that
. What can you say about
, if m and n are coprime?
Let . Show that G is the cyclic group of order six.
Consider the natural action of on
, the set of
real matrices, by left multiplication.
(i) Under this action, if , show that the stabiliser of
is {I}, where I is the 2 × 2 identity matrix.
(ii) Suppose that det(x) = 0 in the remaining parts of this exercise. We will show that the stabiliser of x is infinite. If x = 0, the stabiliser of x is . So suppose x ≠ 0. Let us write
. Then,
for non-zero
. Why ?
(iii) Let be a vector that is not a scalar multiple of
. Show that there is a matrix
such that
and
.(Hint: Set up two sets of simultaneous equations in two unknowns and argue why they have a solution.)
(iv) Check that I−b is in the stabiliser of x. Also, show that there are infinitely many choices of for which I − b is invertible.