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.
Let (J=\begin{bmatrix}0&I\-I&0\end{bmatrix}). By definition, a (2n\times 2n) matrix (M) is symplectic if and only if [ M^{t}JM=J. ] Write [ M=\begin{bmatrix}A&B\ C&D\end{bmatrix},\qquad M^{t}=\begin{bmatrix}A^{t}&C^{t}\ B^{t}&D^{t}\end{bmatrix}. _____ ____ __________ _______ _______ ______.
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Suppose n is even, , where
. Let
and
. Show that HN is a subgroup of Dn. What is its order?
) Let be a finite extension F of odd degree (greater than 1). Show that
Find all the Sylow 2-subgroups of Dn when n is odd. Describe them in terms of x and y.
Suppose n is as in the previous part. Find all the Sylow 2-subgroups of Dn. Describe them in terms of x and y.
If , show that a polynomial ax2 + bx + c is irreducible iff
where
is the group of squares in
Let H be a finite group and, for some prime p, let P be a p-Sylow subgroup of H which is normal in H. Suppose H is normal in K, where K is a finite group. Then, show that P is normal in K.
In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group