P-complement

Definition
Let $$G$$ be a finite group and $$p$$ be a prime number. A p-complement (sometimes called a $$p$$-Sylow complement) in $$G$$ can be defined in the following equiavlent ways:


 * It is a subgroup of $$G$$ whose order is relatively prime to $$p$$ and whose index is a power of $$p$$.
 * It is a subgroup whose order is the largest divisor of the order of $$G$$ that is relatively prime to $$p$$.
 * It is a $$p'$$-Hall subgroup, i.e., a Hall subgroup in $$G$$ for the set of all primes excluding $$p$$.
 * It is a permutable complement to any $$p$$-Sylow subgroup of $$G$$.

Note that $$p$$-complements need not exist. It is also possible for a group to have more than one conjugacy class of $$p$$-complements. In fact, Hall's theorem shows that if $$p$$-complements exist in a finite group for all primes $$p$$, then the group is a finite solvable group.

Particular cases

 * If $$p$$ does not divide the order of $$G$$, then the whole group $$G$$ itself is the unique $$p$$-complement.
 * If $$G$$ is a finite p-group, then the trivial subgroup is the unique $$p$$-complement.
 * There is a unique $$p$$-complement if and only if it is a normal p-complement.