I think you are getting a tad confused with order and index. The theorem from your notes should, I believe, be saying that every subgroup of index 2
is normal. A subgroup has index if , where is, as I'm sure you know, the order (aka number of elements) of .
Therefore, you have the first result, as the subgroup you are given has index two, and so is normal.
For the second result, which groups of order 6 do you know? There is one abelian one, but you can forget about this as every subgroup of an abelian group is normal. So, take your non-abelian group of order 6. Take an element of order 2 in it. Does this element generate a normal subgroup?