- Study: 14.2, 14.3
- Do: p562: 7 (with x^4 - 2 instead of x^8 - 2; note we will do the full Galois correspondence for this polynomial on Monday or Wednesday),
8 (you may want to look up a bit on "p-groups"),
9 (this should be easy),
11 The second part requires a proof or counterexample.
To do the first part, do the following exercise:
Consider the symmetric group S_4.
It has a normal subgroup, A_4, of index 2.
Show that A_4 is the only subgroup of S_4 of index 2.
Hint: Suppose K is another such subgroup. Note that it too is normal.
Show that S_4 = K A_4 and that the intersection of A_4 and K has
index 2 in A_4 (subhint: second isomorphism theorem).
The diagram on page 112 may also prove useful.