Normal subgroups in groups of odd order I put the following question in my first-year algebra final this year: Suppose $G$ is a finite group of odd order and $N$ is a normal subgroup of order $5$. Show that $N\le Z(G)$. (By the way, this problem has been posed on this site before.)
The proof that I guided them through goes like this: all the conjugacy classes of $G$ have odd order; since $N$ is normal, it is a union of conjugacy classes. The only possibilities are $3$, $1$, $1$, and five $1$'s. In either case, $N$ contains a nonidentity element whose conjugacy class consists only of itself, so it is in $Z(G)$; but that element generates $N\cong \mathbb{Z}/5$ and the result follows. 
So let $S(p)$ be the following statement: If $G$ is a finite group of odd order and $N$ is a normal subgroup of order $p$, then $N\le Z(G)$. For which (odd) $p$ does this hold? The argument above shows that it holds for $p=5$, but pretty clearly that proof will not work for $p>5$. 
In fact, if $p$ is any prime that is not a Fermat prime, $S(p)$ is false; a counterexample follows. 
Let $q$ be an odd prime dividing $p-1$, and consider the nonabelian group $G$ of order $pq$. It must have only one $p$-Sylow subgroup, since the number of such subgroups divides $q$ is and $\equiv 1\mod{p}$, so it is normal. So $G$ satisfies the conditions of the theorem. But the center of $G$ is trivial since otherwise $G/Z(G)$ is cyclic and thus $G$ would be abelian. This counterexample does not work when $q=2$ since the group of order $pq$ has even order.
So my question is: does $S(p)$ hold when $p$ is a Fermat prime?
 A: Yes this holds for Fermat primes. You need the following observation. This is sometimes called the $N/C$ Theorem. 
Lemma Let $G$ be a group with a subgroup $H$, then $N_G(H)/C_G(H)$ embeds homomorphically into Aut$(H)$.
Here $N_G(H)=\{g \in G : gH=Hg\}$, the normalizer of $H$ in $G$ and $C_G(H)=\{g \in G : gh=hg \text { for all } h\in H\}$, the centralizer of $H$ in $G$.
Now let us apply this to the situation where $N=H$ is normal, $|N|=$ Fermat-prime and $|G|$ is odd. Then $N_G(N)=G$ because of the normality of $N$. And since $N$ is cyclic,  |Aut$(N)|$ is a power of $2$ (in fact $|N|-1$). It follows from the $N/C$ theorem that $|G/C_G(N)|$ is a power of $2$. But obviously it also divides $|G|$, which is odd. This can only be when $G=C_G(N)$, that is $N \subseteq Z(G)$.
So how do you prove the lemma? Let me give a sketch and leave the details with you: $N_G(H)$ acts as automorphisms on $H$ by conjugation. The kernel of the action is $C_G(H)$.
Generalizations:
Proposition 1 Let $G$ be a finite group, $N$ a normal subgroup of prime order $p$, with gcd$(|G|,p-1)=1$, then $N \subseteq Z(G)$.
Note that this holds for $p=2$: a normal subgroup of order $2$ must be central.
Proposition 2 Let $G$ be a finite group, $N$ a normal subgroup of prime order $p$, where $p$ is the smallest prime dividing $|G|$. Then $N \subseteq Z(G)$.
A: Let $N$ be a cyclic group of prime order, and let $A$ be any group of automorphisms of $N$. Then the orbits of $N\setminus\{e\}$ under $A$ all have the same size. (The proof is easy, based on the fact that every element of $N\setminus\{e\}$ is a generator of $N$, and hence every element of $N\setminus\{e\}$ is a power of every other element.) Note that this holds for your counterexample: all the orbits have size $q$.
In particular, if $N$ is a normal subgroup of $G$, then taking $A$ to be $G$ acting by conjugation, we see that all conjugacy classes in $N\setminus\{e\}$ have the same size. Furthermore, if $N$ has odd order, then all these conjugacy classes have odd size. Finally, if the order of $N$ is a Fermat prime, then the only odd divisor of $\#N-1$ is 1. Therefore every element of $N$ is its own conjugacy class, hence is in the center.
(PS: No way I could have answered this question if you hadn't written it so well!)
A: In addition to the things I answered here, I just want to add the dual of Proposition 1: 
Proposition 1* Let $G$ be a finite group and $H$ a subgroup of prime index $p$, with gcd$(|G|,p-1)=1$. Then $G' \subseteq H$.
 Note that this implies that $H \unlhd G$, and that it is in fact sufficient to prove that $H$ is normal, since then $G/H \cong C_p$ is abelian.
Proof  This is more sophisticated than the proof below: firstly, we may assume by induction on $|G|$, that $H$ is core-free, that is core$_G(H)=\bigcap_{g \in G}H^g=1$. This means that $G$ can be homomorphically embedded in $S_p$. Let $P \in Syl_p(G)$ and note that because $|S_p|=p \cdot (p-1) \cdots \cdot 1$, $|P|=p$. By the $N/C$-Theorem, $N_G(P)/C_G(P)$ embeds in Aut$(P) \cong C_{p-1}$. By the assumption gcd$(|G|,p-1)=1$, we get that $N_G(P)=C_G(P)$. Since $P$ is abelian we have $P \subseteq C_G(P)$, whence $P \subseteq Z(N_G(P))$. We now can apply Burnside's Normal $p$-complement Theorem, which implies that $P$ has a normal complement $N$, that is $G=PN$ and $P \cap N=1$. Note that $|G/N|=p$. Look at the image of $H$ in $G/N$. Then $G=HN$, or $HN=N$. In the latter case $H \subseteq N$, and $|G:H|=|G:N|=p$, whence $H=N$ and we are done if we can refute the first case. If $G=HN$, then $|G:H \cap N|=|G:N|\cdot|N:H \cap N|=|G:N|\cdot |G:H|=p \cdot p=p^2$, contradicting the fact that $|G| \mid |S_p|$. The proof is now complete.
Corollary 1 Let $G$ be a finite group and let $H$ be a subgroup with $|G:H|=p$, the smallest prime dividing the order of $G$. Then $G' \subseteq H$. In particular, $H$ is normal.
Corollary 2 Let $G$ be a finite group and let $H$ be a subgroup with $|G:H|=p$ and gcd$(|H|,p-1)=1$. Then $H$ is normal.
Observe that this last result renders a well-known result for $p=2$! Finally for fun:
Corollary 3 Let $G$ be a finite group of odd order and $H$ a subgroup with $|G:H|=65537$. Then $H$ is normal.
