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.