# P-complement

From Groupprops

## Definition

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

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

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

## Particular cases

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