Normal p-complement

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The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter
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Definition

Symbol-free definition

A subgroup of a group is said to be a normal p-complement if it is a normal subgroup and it has a Sylow subgroup (particularly a Sylow p-subgroup) as a permutable complement.

A group that has a normal p-complement is termed a p-nilpotent group.

Definition with symbols

A subgroup H of a group G is said to be a normal p-complement if it satisfies the following equivalent conditions:

  1. H is normal in G and there is a Sylow p-subgroup P of G such that HP = G and H \cap P is trivial.
  2. H is normal in G and for every Sylow p-subgroup P of G, HP = G and H \cap P is trivial.
  3. H is a normal Hall subgroup of G whose order is relatively prime to p and whose index is a power of p. In other words, H is a normal p'-Hall subgroup of G.

If G contains a normal p-complement H, we say that G is a p-nilpotent group.

Facts

Normal p-complements may not always exist.

A complete list of normal p-complement theorems is available at:

Category:Normal p-complement theorems