Normal subgroups

Equivalent definitions

For any group $G$ :

The center $Z (G)$ is always normal in $G$ ; it is proper exactly when $G$ is nonabelian.
If there is a positive integer $d$ such that $G$ has exactly one subgroup of order $d$ , then that subgroup must be normal; in fact, it must be characteristic.

For any finite group $G$ :

Any subgroup with index equal to the smallest prime divisor of $| G |$ is normal. In other words, if the smallest prime divisor of $|$ G| is $p$ , then any subgroup $H \leq G$ with $[G : H] = p$ is normal. Such a subgroup is always proper; it is nontrivial exactly when $| G | > p$ .
- Special case: Any subgroup of index 2 is always normal. (Of course, this is only possible in a group of even order.)
Let $p$ be any prime divisor of $|$ G|, and write $| G | = p^{m} d$ with $gcd (d, p) = 1$ . Then the Sylow $p$ -subgroups of $G$ are the subgroups of order $p^{m}$ . Let $n_{p}$ denote the number of such subgroups. The Sylow Theorem guarantees that:
- $n_{p} \geq 1$ , i.e., there exists at least one Sylow $p$ -subgroup
- $n_{p}$ divides $d$
- $n_{p} \equiv 1 (\mod p)$
- The Sylow $p$ -subgroups are all conjugate to each other
  Note that if $n_{p} = 1$ then the unique Sylow $p$ -subgroup is normal (characteristic, even); if $n_{p} > 1$ then the Sylow $p$ -subgroups are not normal. Also note that Sylow $p$ -subgroups are always nontrivial, and they are proper unless $d = 1$ , i.e., $G$ is a $p$ -group itself.