P-nilpotent group

From Groupprops
Jump to: navigation, search
The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties

Definition

A finite group G is termed a p-nilpotent group for a prime number p if the following equivalent conditions are satisfied:

  1. G has a normal p-complement, i.e., a normal Hall subgroup whose order is coprime to p and whose index is a power of p.
  2. The p-Sylow subgroups of G are retracts of G.
  3. O_{p',p}(G) = G.