Find where
.
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In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group
Suppose n is even, , where
. Let
and
. Show that HN is a subgroup of Dn. What is its order?
The aim of this exercise is to show that acts transitively on
Check that is in the stabiliser of
. Also, show that there are infinitely many choices of
for which
is invertible.
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 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 ?
Consider the natural action of on
, the set of
real matrices, by left multiplication.