Find the elementary divisors and invariant factors of .
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Let . Show that G is the cyclic group of order six.
(b) Solve the following set of congruences:
(c) Show that is not a UFD by giving two different factorisations of 20.
Suppose n is as in the previous part. Find all the Sylow 2-subgroups of Dn. Describe them in terms of x and y.
Let p be an odd prime that divides . Suppose
. Show that C is the unique Sylow p-subgroup of Dn.
Let and let
be algebraic over F of degree m and n, respectively. Show that
. What can you say about
if m and n are coprime?
In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group
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.