P-complement

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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.