Normal Hall subgroup

Symbol-free definition
A subgroup of a finite group is said to be a normal Hall subgroup if it satisfies the following equivalent conditions:


 * 1) It is a Hall subgroup, viz., the order and index are relatively prime, and is a normal subgroup viz., every inner automorphism of the whole group takes the subgroup to itself,
 * 2) It is a Hall subgroup and is a characteristic subgroup: every automorphism of the group takes the subgroup to itself.
 * 3) It is a Hall subgroup and is a fully characteristic subgroup: every endomorphism of the group takes the subgroup to itself.
 * 4) It is a Hall subgroup and is a subnormal subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be a normal Hall subgroup if it satisfies the following equivalent conditions:


 * $$|H|$$ and $$[G:H]$$ are relatively prime, and $$H \triangleleft G$$, viz., $$gHg^{-1} \leq H$$ for any $$g \in G$$
 * $$|H|$$ and $$[G:H]$$ are relatively prime, and $$H \operatorname{char} G$$, viz., $$\sigma(H) \le H$$ for all $$\sigma \in \operatorname{Aut}(G)$$
 * $$|H|$$ and $$[G:H]$$ are relatively prime, and $$H \operatorname{char} G$$, viz., $$\sigma(H) \le H$$ for all $$\sigma \in \operatorname{End}(G)$$
 * $$|H|$$ and $$[G:H]$$ are relatively prime, and $$H$$ is subnormal in $$G$$.

Equivalence of definitions
Check out: Hall implies intermediately normal-to-characteristic, Hall implies intermediately subnormal-to-normal.

Stronger properties

 * Weaker than::Hall direct factor
 * Weaker than::Normal Sylow subgroup

Weaker properties

 * Stronger than::Order-unique subgroup
 * Stronger than::Permutably complemented normal subgroup:
 * Stronger than::Permutably complemented subgroup:
 * Stronger than::Homomorph-containing subgroup
 * Stronger than::Fully characteristic subgroup:
 * Stronger than::Characteristic subgroup

Metaproperties
The property of being a normal Hall subgroup is characteristic because:


 * The Hall part is transitive
 * The normal part becomes transitive because the property of being a Hall subgroup is a transitivizer of normality, or more specifically, because it is a normal-to-characteristic subgroup property, and the subgroup proeprty of being characteristic is transitive.

Since both the properties of being normal and of being Hall satisfy the transfer condition, so does the property of being a normal Hall subgroup.