The aim of this exercise is to show that acts transitively on
_________ _______ ____ ______ __________ _________ ____ __________ ______ ________ _________.
___ ______ __________ _________ _________ __________ ___ ___ _______ ___ ________.
______ _____ _______ _______ _______ ________ _____ ______.
___ ____ ________ ___ ________.
_____ ________ _______ _____ ________ _____ _____ ___.
________ _______ _________ ___ __________ ____ _______ _________.
___ ______ __________ _______ ___ _______ ___ _______ ________ ______.
______ _________ _____ ________ _________ ____.
_________ ________ _______ __________ ________ ____.
_________ ____ _________ _______ _________ ____ ________ ____ _________ _____ ________.
_________ _________ ___ ______ ______ _________ ____ ____ ____ _________.
_________ _______ ___ ____ __________ ___ ____ ____ __________ ___ ____ _______.
________ ________ _____ _________ _______ __________ ______.
__________ ____ __________ ______ _________ __________ _____ _________.
____ ________ ______ _____ _________ ___.
_________ __________ _________ __________ _________ ______ ___.
________ ________ _________ _______ ______ ____ ________ ______ _________ __________ _________.
___ ______ __________ ________ ________ __________ ________ _____ _________ _______ ______.
_________ _____ __________ ______ __________ ___.
______ ______ ____ _________ ________ ____ _______.
____ _______ _________ ____ _____ _________ _____ ___ ________ _____.
______ _________ ______ ___ ___ __________.
____ ____ _____ ________ ____ ________ _______.
____ _____ __________ ______ ______ ______ ________ ______ ______ _________ _________ ____.
______ _________ _____ ____ ________ _____ _______ _______.
____ _____ ____ ________ __________ _______ ______ __________ _____ _______ ________.
________ _____ ____ ________ ______ _________ ______ _____ __________ ___ ____.
____ ________ ______ _____ _____ _______ _________ ________ _________.
_____ _____ ________ ________ _____ ___ _______ _______ _____.
___ ______ ________ __________ ________ ___ __________ ________.
______ _____ ___ ________ _________ ____ _______ ______.
- Solved Assignments
- Study Notes
- Guess Papers
- Question Papers
✓ Unlock up to 60% of the answer
✓ No payment required
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.)
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 finite extension F of odd degree (greater than 1). Show that
Suppose n is as in the previous part. Find all the Sylow 2-subgroups of Dn. Describe them in terms of x and y.
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.
) Prove the relation
Further, find all the elements of order 2 in Dn.
Suppose n is even, , where
. Let
and
. Show that HN is a subgroup of Dn. What is its order?