Sylow's theorem

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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

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

Similar statements

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