Consider the natural action of on
, the set of
real matrices, by left multiplication.
Define the action of (GL_2(\mathbb{R})) on (\mathbf{M}_2(\mathbb{R})) by [ A \cdot X = AX,\quad A \in GL_2(\mathbb{R}),\ X \in \mathbf{M}_2(\mathbb{R}). ] This is a well-defined group action _________ ___ _________ ____ ________ _____ __________ ____ ____ _________.
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Suppose n is even, , where
. Let
and
. Show that HN is a subgroup of Dn. What is its order?
Find all the Sylow 2-subgroups of Dn when n is odd. Describe them in terms of x and y.
) Prove the relation
Further, find all the elements of order 2 in Dn.
If is a finite field show that there is always an irreducible polynomial of the form x3 - x + a where
. (Hint: Show that
is not a surjective map.)
Show that (i) Show that a matrix
is symplectic if and only if
.
(ii) Show that, to prove that
acts transitively on
, 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
?)
(iii) Complete the proof by showing that, given any non-zero vector
, there is always a non-zero vector
such that
is symplectic.
Suppose that in the remaining parts of this exercise. We will show that the stabiliser of
is infinite. If
, the stabiliser of
is
. So suppose
. Let us write
. Then,
for non-zero
. Why ?
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.)