Consider the natural action of on
, the set of
real matrices, by left multiplication.
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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.
If is a finite field show that there is always an irreducible polynomial of the form x3 - x + a where
. (Hint: Show that
is not a surjective map.)
In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group
By looking at the factorisation of guess the number of irreducible polynomials of degree 2 over
. Find all the irreducible polynomials of degree 2 over
.
If , show that a polynomial ax2 + bx + c is irreducible iff
where
is the group of squares in
Check that is in the stabiliser of
. Also, show that there are infinitely many choices of
for which
is invertible.
Consider the natural action of on
, the set of
real matrices, by left multiplication.