Sylow's theorem

This article gives the statement, and possibly proof, of a basic fact in group theory.
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This fact is related to: Sylow 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

• ECD condition for pi-subgroups in solvable groups: In a finite solvable group, an analogue of Sylow's theorem holds for sets of primes in place of primes, with Sylow subgroups being replaced by Hall subgroups. Existence, conjugacy and domination have obvious analogues here. There are analogues to the divisibility and congruence conditions as well, but these are more complicated.
• Sylow's theorem in profinite groups
• Sylow's theorem with operators: An analogue of Sylow's theorem where, instead of looking at all $p$-subgroups, we consider the $p$-subgroups invariant under the action of a coprime automorphism group. The known proofs of this invoke the odd-order theorem, in the guise of the fact that given two groups of coprime order, one of them is solvable.

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}