ECD condition

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

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.

Examples

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.