Compute the following:
(i) (ii)
The numerical values of the ______ _______ ______ _______ ___ ___ ____ ______ __________ _____.
________ ___ ________ _____ _______.
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Consider the natural action of GL2 on M2
, the set of 2 x 2 real matrices, by left multiplication.
(i) Under this action, if det(x) ≠ 0, show that the stabiliser of x ∈ M2 is {I}, where I is the 2 x 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 GL2. So suppose x ≠ 0. Let us write
Then,
for non-zero λ ∈ R. Why?
(iii) Let be a vector that is not a scalar multiple of
. Show that there is a matrix b =
such that b
= 0 and b
= α
(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 a for which I - b is invertible.
Let . Show that G is the cyclic group of order six.
Prove the relation
Further, find all the elements of order 2 in Dn.
Find the elementary divisors and invariant factors of
The aim of this exercise is to show that SP2 acts transitively on
\ {0}.
Solve the following set of congruences:
Let p be an odd prime that divides n, n = pr l, p + 1. Suppose C = (x1). Show that C is the unique Sylow p-subgroup of Dn.