# Normal subgroups

# Equivalent definitions

# Existence of normal subgroups

For any group

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

For any finite group

- Any subgroup with index equal to the smallest prime divisor of
is normal. In other words, if the smallest prime divisor of G| is , then any subgroup with is normal. Such a subgroup is always proper; it is nontrivial exactly when . - Special case: Any subgroup of index 2 is always normal. (Of course, this is only possible in a group of even order.)

- Let
be any prime divisor of G|, and write with . Then the Sylow -subgroups of are the subgroups of order . Let denote the number of such subgroups. The Sylow Theorem guarantees that: , i.e., there exists at least one Sylow -subgroup divides - The Sylow
-subgroups are all conjugate to each other

Note that ifthen the unique Sylow -subgroup is normal (characteristic, even); if then the Sylow -subgroups are not normal. Also note that Sylow -subgroups are always nontrivial, and they are proper unless , i.e., is a -group itself.