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{{subgroup metaproperty}}

==Definition==

===Definition with symbols===

A [[subgroup property]] <math>p</math> is said to satisfy the '''ECD condition''' if the following are true:

* '''Existence (E)''': Every group has a subgroup satisfying <math>p</math> (this is the same as being [[right-realized subgroup property|right-realized]]
* '''Domination (D)''': Every subgroup with property <math>p</math> is contained in a subgroup maximal with respect to having the property <math>p</math>.
* '''Conjugacy (C)''': Any two subgroups maximal with respect to having the property <math>p</math> 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 <math>p</math> and a group property <math>q</math>, we say that <math>p</math> satisfied the ECD condition for groups with property <math>q</math>, if in groups with property <math>q</math>, the subgroup property corresponding to <math>p</math> 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.

ECD condition - Revision history

Vipul: 1 revision

Vipul at 07:47, 22 February 2007