Sylow subgroup

Equivalent definitions in tabular format
A subgroup of a finite group is termed a Sylow subgroup if it is a $$p$$-Sylow subgroup for some prime number $$p$$. We give equivalent definitions of a $$p$$-Sylow subgroup.

Note that the trivial subgroup is always a Sylow subgroup: it is $$p$$-Sylow for any prime $$p$$ not dividing the order of the group. The whole group is $$p$$-Sylow as a subgroup of itself if and only if it is a $$p$$-group.

Equivalence of definitions
Note that the equivalence of definitions (1)-(3) is a simple exercise in Lagrange's theorem and basic facts about prime factorization. The equivalence with definition (4) relies on the fact that Sylow subgroups exist and Sylow implies order-dominating, which are both considered parts of Sylow's theorem.

ECD condition
The property of being a $$p$$-Sylow subgroup is obtained as the property of being maximal corresponding to the group property of being a $$p$$-Sylow subgroup. It turns out that:


 * Existence (E): For every $$p$$, there exist $$p$$-Sylow subgroups.
 * Domination(D): Any $$p$$-group is contained in a $$p$$-Sylow subgroup.
 * Conjugacy(C): Any two $$p$$-Sylow subgroups are conjugate.

All these facts, together, show that the group property of being a $$p$$-group satisfies the ECD condition.

Weaker properties
For a better understanding of how all these facts about Sylow subgroups are proved, refer the survey articles deducing basic facts about Sylow subgroups and Hall subgroups and deducing advanced facts about Sylow subgroups and Hall subgroups

Textbook references

 * , Page 206, Point (4.5) (formal definition)