Sylow subgroup

Definition

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.

No.	Shorthand	A subgroup of a group is a $p$-Sylow subgroup if...	A subgroup $H$ of a group $G$ is a $p$-Sylow subgroup of $G$ if ...	Applications to...	Additional comments
1	order prime, index coprime	it is a $p$-group (i.e., its order is a power of $p$, so it is a group of prime power order) and its index is relatively prime to $p$	$H$ is a $p$-group (i.e., its order is $p^r$ for some nonnegative integer $r$), so it is a group of prime power order) and its index $[G:P]$ is relatively prime to $p$.
2	order is maximal power of prime definition	the order of the subgroup is the largest power of $p$ dividing the order of the group	the order of $H$ is the largest power of $p$ dividing the order of $G$. In other words, if the order of $G$ is $p^rm$ where $r$ is a nonnegative integer and $m$ is an integer relatively prime to $p$, $H$ must have order $p^r$.
3	Hall subgroup plus p-subgroup definition	it is a $p$-group (nontrivial if $p$ divides the group order) and also a Hall subgroup: its order and index in the whole group are relatively prime	it is a $p$-group (nontrivial if $p$ divides the order of $G$) and also a Hall subgroup: its order $|H|$ and index $[G:H]$ are relatively prime.
4	maximal element among p-subgroups	it is a $p$-subgroup of the whole group and is not contained in any bigger $p$-subgroup	$H$ is a $p$-subgroup of $G$ (i.e., the order of $H$ is a power of $p$) and any $p$-subgroup of $G$ containing $H$ must equal $H$

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.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter
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This article defines a subgroup property that makes sense within a finite group

This article describes a property that arises as the conjunction of a subgroup property: Hall subgroup with a group property (itself viewed as a subgroup property): group of prime power order
View a complete list of such conjunctions

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. For full proof, refer: Sylow subgroups exist
• Domination(D): Any $p$-group is contained in a $p$-Sylow subgroup. For full proof, refer: Sylow implies order-dominating
• Conjugacy(C): Any two $p$-Sylow subgroups are conjugate. For full proof, refer: Sylow implies order-conjugate

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

Relation with other properties

Conjunction with other properties

Conjunction	Other component of conjunction	Intermediate notions between Sylow subgroup and conjunction	Intermediate notions between other component and conjunction	Additional comments
normal Sylow subgroup	normal subgroup	|FULL LIST, MORE INFO	Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Complemented normal subgroup, Fully invariant subgroup, Homomorph-containing subgroup, Isomorph-free subgroup, Normal subgroup having no common composition factor with its quotient group, Normal subgroup having no nontrivial homomorphism from its quotient group, Normal subgroup having no nontrivial homomorphism to its quotient group, Normal-homomorph-containing subgroup, Order-normal subgroup, Order-unique subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO	a normal Sylow subgroup is automatically a characteristic subgroup. In fact, it is verbal and variety-containing. Further, by the Schur-Zassenhaus theorem, it is automatically a complemented normal subgroup.
Sylow direct factor	direct factor	|FULL LIST, MORE INFO	Fully invariant direct factor, Hall direct factor|FULL LIST, MORE INFO	Any Sylow subgroup that is a central factor is automatically a direct factor.
Sylow retract	retract	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO	any Sylow subgroup that is a conjugacy-closed subgroup is automatically a Sylow retract.
abelian Sylow subgroup	abelian subgroup (via group property abelian group)	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
order-dominating subgroup	any subgroup of the whole group whose order divides the order of the Sylow subgroup is contained in a conjugate of the Sylow subgroup	Sylow implies order-dominating		|FULL LIST, MORE INFO
order-dominated subgroup	any subgroup of the whole group whose order is a multiple of that of the Sylow subgroup contains a conjugate of the Sylow subgroup	Sylow implies order-dominated		|FULL LIST, MORE INFO
order-conjugate subgroup	any other subgroup of the group of the same order is conjugate to it	Sylow implies order-conjugate		Order-dominated subgroup, Order-dominating subgroup|FULL LIST, MORE INFO
order-automorphic subgroup	any other subgroup of the group of the same order is automorphic to it	(also via order-conjugate)		Order-conjugate subgroup|FULL LIST, MORE INFO
order-isomorphic subgroup	any other subgroup of the group of the same order is isomorphic to it	(via order-conjugate)		Order-conjugate subgroup|FULL LIST, MORE INFO
isomorph-automorphic subgroup	any other subgroup isomorphic to it is automorphic to it	(via order-conjugate), see Sylow implies isomorph-automorphic		Order-conjugate subgroup|FULL LIST, MORE INFO
isomorph-conjugate subgroup	any other subgroup isomorphic to it is conjugate to it	(via order-conjugate), see Sylow implies isomorph-conjugate		Intermediately isomorph-conjugate subgroup, Nilpotent Hall subgroup, Order-conjugate subgroup, Order-dominated subgroup|FULL LIST, MORE INFO
automorph-conjugate subgroup	any other subgroup automorphic to it is conjugate to it	(via order-conjugate), see Sylow implies automorph-conjugate		Characteristic subgroup of Sylow subgroup, Intermediately automorph-conjugate subgroup, Order-conjugate subgroup|FULL LIST, MORE INFO
intermediately isomorph-conjugate subgroup	isomorph-conjugate in every intermediate subgroup	Sylow implies intermediately isomorph-conjugate		Intermediately isomorph-conjugate subgroup, Nilpotent Hall subgroup, Order-conjugate subgroup, Order-dominated subgroup|FULL LIST, MORE INFO
intermediately automorph-conjugate subgroup	automorph-conjugate in every intermediate subgroup	Sylow implies intermediately automorph-conjugate		Intermediately isomorph-conjugate subgroup|FULL LIST, MORE INFO
Hall subgroup	order and index are relatively prime	(by definition)	whole group gives counterexample when it isn't a $p$-group	Nilpotent Hall subgroup, Pronormal Hall subgroup|FULL LIST, MORE INFO
nilpotent Hall subgroup	Hall and also nilpotent	prime power order implies nilpotent	whole group gives counterexample when it is nilpotent and not a $p$-group	|FULL LIST, MORE INFO
Order-dominating Hall subgroup	Hall and order-dominating subgroup	(via order-dominating + Hall)		|FULL LIST, MORE INFO
procharacteristic subgroup		Sylow implies procharacteristic		Nilpotent Hall subgroup|FULL LIST, MORE INFO
weakly procharacteristic subgroup		(via procharacteristic)		|FULL LIST, MORE INFO
paracharacteristic subgroup		(via procharacteristic)		Hall subgroup, Join of Sylow subgroups|FULL LIST, MORE INFO
polycharacteristic subgroup		(via procharacteristic)		Hall subgroup, Join of Sylow subgroups|FULL LIST, MORE INFO
weakly characteristic subgroup		(via procharacteristic)		|FULL LIST, MORE INFO
intermediately normal-to-characteristic subgroup	characteristic subgroup inside any intermediate subgroup in which it is normal			Hall subgroup, Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup|FULL LIST, MORE INFO
pronormal subgroup	any conjugate subgroup is conjugate to it in their join	Sylow implies pronormal		Intermediately isomorph-conjugate subgroup, Nilpotent Hall subgroup, Nilpotent pronormal subgroup, Sylow subgroup of normal subgroup|FULL LIST, MORE INFO
weakly pronormal subgroup		(via pronormal)		Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup, Pronormal subgroup|FULL LIST, MORE INFO
paranormal subgroup		(via pronormal)		Hall subgroup, Intermediately isomorph-conjugate subgroup, Join of Sylow subgroups, Pronormal subgroup|FULL LIST, MORE INFO
polynormal subgroup		(via pronormal)		Hall subgroup, Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup, Join of Sylow subgroups, Paranormal subgroup, Weakly pronormal subgroup|FULL LIST, MORE INFO
weakly normal subgroup		(via pronormal)		Paranormal subgroup|FULL LIST, MORE INFO
intermediately subnormal-to-normal subgroup		(via pronormal)		Hall subgroup, Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup, Intermediately normal-to-characteristic subgroup, Paranormal subgroup, Polynormal subgroup, Pronormal subgroup, Weakly pronormal subgroup|FULL LIST, MORE INFO
subgroup with abnormal normalizer		(via pronormal)		Pronormal subgroup|FULL LIST, MORE INFO
subgroup with weakly abnormal normalizer		(via weakly pronormal)		|FULL LIST, MORE INFO
subgroup with self-normalizing normalizer		(via pronormal, weakly pronormal)		Pronormal subgroup, Weakly pronormal subgroup|FULL LIST, MORE INFO
Closure-characteristic subgroup	normal closure is a characteristic subgroup	(via automorph-conjugate, also via Hall), see also Sylow implies closure-characteristic		Automorph-conjugate subgroup, Hall subgroup, Join of Sylow subgroups, Subgroup whose normal closure is fully invariant, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
Core-characteristic subgroup	normal core is a characteristic subgroup	(via automorph-conjugate, also via Hall), see also Sylow implies core-characteristic		Automorph-conjugate subgroup, Hall subgroup, Intersection of Sylow subgroups, Subgroup whose normal core is fully invariant|FULL LIST, MORE INFO
WNSCDIN-subgroup		(via pronormal), see also Sylow implies WNSCDIN		Pronormal subgroup|FULL LIST, MORE INFO
MWNSCDIN-subgroup		(via pronormal), see also Sylow implies MWNSCDIN		Pronormal subgroup|FULL LIST, MORE INFO
subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed		(via WNSCDIN)		Pronormal subgroup|FULL LIST, MORE INFO
intermediately normal-to-complemented subgroup	complemented normal subgroup in its normalizer	Schur-Zassenhaus theorem		|FULL LIST, MORE INFO
intermediately central factor-to-direct factor	direct factor in any intermediate subgroup in which it is a central factor			Permuting transfer-closed central factor-to-direct factor, Permuting transfer-closed conjugacy-closed-to-retract|FULL LIST, MORE INFO
intermediately conjugacy-closed-to-retract		Conjugacy-closed and Sylow implies retract		Permuting transfer-closed conjugacy-closed-to-retract|FULL LIST, MORE INFO

Metaproperties

Metaproperty	Satisfied?	Proof	Statement with symbols
intermediate subgroup condition	Yes	Sylow satisfies intermediate subgroup condition	If $H$ is a Sylow subgroup of $G$, and $K$ is any intermediate subgroup of $G$ containing $H$, then $H$ is a Sylow subgroup of $K$.
transfer condition	No	Sylow does not satisfy transfer condition	If $H,K \le G$, with $H$ a Sylow subgroup of $G$, $H \cap K$ need not be a Sylow subgroup of $K$.
permuting transfer condition	Yes	If $H,K \le G$ with $H$ a Sylow subgroup of $G$, and $HK = KH$, then $H \cap K$ is a Sylow subgroup of $K$.
image condition	Yes	If $H$ is a Sylow subgroup of $G$ and $\varphi:G \to K$ is surjective, then $\varphi(H)$ is a Sylow subgroup of $K$	Sylow satisfies image condition

References

Textbook references

• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 206, Point (4.5) (formal definition)