ECD condition

Definition with symbols
A subgroup property $$p$$ is said to satisfy the ECD condition if the following are true:


 * Existence (E): Every group has a subgroup satisfying $$p$$ (this is the same as being right-realized
 * Domination (D): Every subgroup with property $$p$$ is contained in a subgroup maximal with respect to having the property $$p$$.
 * Conjugacy (C): Any two subgroups maximal with respect to having the property $$p$$ are conjugate to each other.

Often, we refer to ECD conditions not for a general subgroup property but for a subgroup property with respect to certain particular groups or with respect to groups with additional structure.

For a pair of group properties
Given a group property $$p$$ and a group property $$q$$, we say that $$p$$ satisfied the ECD condition for groups with property $$q$$, if in groups with property $$q$$, the subgroup property corresponding to $$p$$ satisfies ECD condition.

Here, by subgroup property corresponding to group property we mean the property of being a subgroup, that as an abstract group, satisfies the group property.

Groups of prime power order and Sylow subgroups
In a finite group, the group property of being a group of prime power order for a fixed prime is an ECD-property, and the maximal operator applied to this yields the property of being a Sylow subgroup. The proof of this is the content of Sylow's theorem.