Find the elementary divisors and invariant factors of
To find the elementary divisors and invariant factors of the abelian group \( G = \mathbb{Z}_8 \times \mathbb{Z}_{12} \times \mathbb{Z}_{15} \), follow these steps: --- ### Step 1: Prime Factorization of Orders Each cyclic group in \( G \) has an order: \[ 8 = 2^3, \quad 12 = 2^2 \times ___ ___ _________ ____ ____ _______ ______ ___ ___.
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Find all the Sylow 2-subgroups of Dn, when n is odd. Describe them in terms of x and y.
Find the elementary divisors and invariant factors of
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.
Solve the following set of congruences:
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 non-zero vector, there is always a
such that
is symplectic.
Let ? (?) be a finite extension F of odd degree(greater than 1). Show that ? (?2) = ? (?).
Suppose n is as in the previous part. Find all the Sylow 2-supgroups of Dn. Describe them in terms of x and y.