Sylow's theorem

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

Verbal statement

The Sylow's theorem(s) give(s) information about the existence of $p$-Sylow subgroups of a finite group, as well as the relation among them. More specifically, given a finite group:

• Existence: For any prime $p$, there exists a $p$-Sylow subgroup
• Conjugacy: Any two $p$-Sylow subgroups are conjugate in the whole group
• Domination: Any $p$-subgroup is contained inside some $p$-Sylow subgroup
• Congruence: The number of $p$-Sylow subgroups divides the index of any $p$-Sylow subgroup and is also congruent to $1$ modulo $p$.

Statement with symbols

Let $G$ be a finite group and $p$ a prime. A subgroup of $G$ is termed a $p$-Sylow subgroup if its order is a power of $p$ and its index is relatively prime to $p$. Note that by Lagrange's theorem, the order of a Sylow subgroup is the largest power of $p$ dividing the order of $G$, and thus, it is a multiple of the order of any $p$-subgroup (subgroup whose order is a power of $p$) of $G$.

Then Sylow's theorem states that:

1. Existence: There exists a $p$-Sylow subgroup $P$ of $G$
2. Conjugacy: If $P$ and $Q$ are $p$-Sylow subgroups of $G$ then there exists $g$ in $G$ such that $gPg^{-1}=Q$ (i.e., $P$ and $Q$ are conjugate subgroups)
3. Domination: Let $P$ be a $p$-Sylow subgroup and $Q$ a $p$-group. Then there exists a $g$ in $G$ such that $gQg^{-1}\subseteq P$.
4. Congruence: Let $Syl_{p}(G)$ denote the set of $p$-Sylow subgroups of $G$ and $n_{p}$ denote the cardinality of $Syl_{p}(G)$. Then, $n_{p}\equiv 1\mod p$.

Related facts

Corollaries

• Sylow implies order-conjugate
• Sylow implies order-isomorphic
• Sylow implies isomorph-automorphic
• Sylow implies automorph-conjugate
• Sylow implies isomorph-conjugate
• Sylow implies intermediately isomorph-conjugate
• Sylow implies intermediately automorph-conjugate
• Sylow implies pronormal

There are other corollaries too, many of which factor through these corollaries.

Similar statements

• Sylow's theorem in profinite groups

Proof breakup

1. Existence: For full proof, refer: Sylow subgroups exist
2. Conjugacy: This follows from Domination, though it can also be proved through the methods used to establish Congruence. For full proof, refer: Sylow implies order-conjugate
3. Domination: For full proof, refer: Sylow implies order-dominating
4. Congruence: For full proof, refer: Congruence condition on Sylow numbers

References

Journal references

• Théorèmes sur les groupes de substitutions by Ludwig Sylow, Mathematische Annalen, Vol. 5, (1872), no. 4, 584-594 ^{Springerlink pageArchived local copy}: The original formulation of Sylow's theorem by Sylow. Does not include the congruence condition part of the theorem, which was proved later by Frobenius.^{More info}
• Sur les groupes transitifs dont le degré est le carré d'un nombre premier by Peter Ludwig Mejdell Sylow, Volume 11, Page 201 - 256(Year 1887): A paper by Sylow on transitive group actions, that begins by stating the complete Sylow's theorem.^{Springerlink pageMore info}