# Sylow theorems

The Sylow theorems are three related results about finite groups that guarantee, for every prime power exactly dividing a group's order, the existence, conjugacy, and enumerability of subgroups of that order. They refine Lagrange's theorem, which states only that the order of a subgroup divides the order of the group, and they apply to every finite group, which makes them the standard first tool for analyzing the structure of a finite group.<sup>[1](https://ocw.mit.edu/courses/res-18-011-algebra-i-student-notes-fall-2021/mit18_701f21_full_lec_new.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | If \( p^{m} \) is the largest power of the prime \( p \) dividing \(\lvert G\rvert \), a Sylow \( p \)-subgroup of \( G \) is a subgroup of order \( p^{m} \)<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup> |
| Existence (Sylow I) | Every finite group has at least one Sylow \( p \)-subgroup for each prime factor of its order<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup> |
| Conjugacy (Sylow II) | All Sylow \( p \)-subgroups of a finite group are conjugate to one another<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup> |
| Counting (Sylow III) | The number \( n_{p} \) of Sylow \( p \)-subgroups satisfies \( n_{p} \equiv 1 \pmod{p} \) and \( n_{p} \) divides \(\lvert G\rvert \)<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup> |
| Normality test | A Sylow \( p \)-subgroup is unique if and only if it is a normal subgroup<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup> |
| Origin | Published by Ludvig Sylow in 1872 in *Mathematische Annalen*, volume 5, pages 584–594, in a 10-page paper<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Sylow/)</sup> |

## Statement

Let \( G \) be a finite group and write \(\lvert G\rvert = p^{a} m \), where \( p \) is a prime not dividing \( m \). A Sylow \( p \)-subgroup is a subgroup of order \( p^{a} \), the largest power of \( p \) dividing the group order. The three theorems then read:<sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/MTHM024/gtrev.pdf)</sup>

1. **Existence.** \( G \) contains subgroups of order \( p^{a} \). More generally, \( G \) contains subgroups of order \( p^{k} \) for every \( 0 \le k \le a \).<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup><sup> • </sup><sup>[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup>
2. **Conjugacy.** Any two Sylow \( p \)-subgroups are conjugate in \( G \). The stronger form says that every \( p \)-subgroup of \( G \) is contained in some conjugate of any fixed Sylow \( p \)-subgroup, so the Sylow \( p \)-subgroups are exactly the maximal \( p \)-subgroups.<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup><sup> • </sup><sup>[6](https://mathweb.ucsd.edu/~drogalsk/200ab-coursenotes-full.pdf)</sup>
3. **Counting.** If \( n_{p} \) denotes the number of Sylow \( p \)-subgroups, then \( n_{p} \equiv 1 \pmod{p} \) and \( n_{p} \) divides \( m \), the \( p \)-free part of the group order.<sup>[4](https://webspace.maths.qmul.ac.uk/p.j.cameron/MTHM024/gtrev.pdf)</sup>

The counting statement is often sharpened by the formula \( n_{p} = [G : N_{G}(P)] \), where \( N_{G}(P) \) is the normalizer of a Sylow \( p \)-subgroup \( P \), the largest subgroup of \( G \) in which \( P \) is normal. Since the index divides \(\lvert G\rvert \), this gives the divisibility part of the third theorem directly.<sup>[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup>

## Motivation: the converse of Lagrange's theorem

Lagrange's theorem says that if \( H \) is a subgroup of a finite group \( G \), then \(\lvert H\rvert \) divides \(\lvert G\rvert \). The natural converse asks whether every divisor of \(\lvert G\rvert \) occurs as the order of a subgroup, and the answer is no: the alternating group \( A_{4} \), of order 12, has no subgroup of order 6.<sup>[1](https://ocw.mit.edu/courses/res-18-011-algebra-i-student-notes-fall-2021/mit18_701f21_full_lec_new.pdf)</sup> This is the smallest possible such example, and the full symmetric group \( S_{4} \), of order 24, also has no subgroup of order 6.<sup>[6](https://mathweb.ucsd.edu/~drogalsk/200ab-coursenotes-full.pdf)</sup>

The Sylow theorems recover the converse for prime-power divisors: if \( d \) divides \(\lvert G\rvert \) and \( d = p^{i} \) is a power of a prime, then \( G \) does have a subgroup of order \( d \).<sup>[6](https://mathweb.ucsd.edu/~drogalsk/200ab-coursenotes-full.pdf)</sup> The first step in this direction was Cauchy's theorem, proved before Sylow's work, which guarantees an element, and hence a cyclic subgroup, of order \( p \) whenever \( p \) divides \(\lvert G\rvert \).<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Sylow/)</sup><sup> • </sup><sup>[6](https://mathweb.ucsd.edu/~drogalsk/200ab-coursenotes-full.pdf)</sup>

## Proofs

The standard modern proofs run through group actions, chiefly the conjugation action of \( G \) on its subgroups and the action by left multiplication on cosets, with the orbit-stabilizer theorem doing the counting.<sup>[7](https://ocw.mit.edu/courses/res-18-011-algebra-i-student-notes-fall-2021/mit18_701f21_lect23.pdf)</sup> A key lemma is that for a \( p \)-subgroup \( H \) of \( G \), the index \( [G:H] \) is congruent to \( [N_{G}(H):H] \) modulo \( p \); applying it to a \( p \)-subgroup acting on cosets yields fixed points, which drive both the existence and conjugacy arguments.<sup>[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup>

Several alternative proofs are known. Sylow's own argument takes a maximal \( p \)-subgroup \( P \) and shows that its index \( [G:P] \) is not divisible by \( p \), so \( P \) must have the maximal possible order. Frobenius gave a proof that embeds \( G \) in a symmetric group by Cayley's theorem and extracts a Sylow subgroup of \( G \) as an intersection with a Sylow subgroup of the symmetric group, using Cauchy's earlier result for symmetric groups.<sup>[8](https://kconrad.math.uconn.edu/blurbs/grouptheory/sylowmore.pdf)</sup> The widely used proof built on counting subsets of \( G \) of size \( p^{a} \) under right multiplication is based on one published by [Helmut Wielandt](https://www.edgepedia.org/helmut-wielandt) in 1959.<sup>[9](https://proofwiki.org/wiki/Sylow_Theorems/Historical_Note)</sup>

Sylow's original paper went beyond the three headline statements. He proved that if \( p^{d} \) divides \(\lvert G\rvert \) then \( G \) has a subgroup of size \( p^{d} \), and that each subgroup of size \( p^{d-1} \) has index \( p \) in a larger subgroup, which produces a nested chain of \( p \)-subgroups from the identity up to a Sylow subgroup. The conjugacy statement, however, does not extend to non-maximal \( p \)-subgroups: subgroups of a fixed non-maximal \( p \)-power order need not be conjugate.<sup>[8](https://kconrad.math.uconn.edu/blurbs/grouptheory/sylowmore.pdf)</sup>

## Applications

The theorems are used to classify groups of a given order and to show that certain orders admit no simple group.<sup>[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup> Two standard techniques are to inspect the divisors of \( m \) and show that \( n_{p} = 1 \) is the only possibility compatible with \( n_{p} \equiv 1 \pmod{p} \), forcing a normal Sylow subgroup, or to count elements of \( p \)-power order across several primes and derive a contradiction.<sup>[10](https://people.math.binghamton.edu/mazur/teach/50324/sylow.pdf)</sup>

**Groups of order \( pq \).** For primes \( p < q \), the number of Sylow \( q \)-subgroups divides \( p \) and is congruent to 1 modulo \( q \), so it must be 1; the Sylow \( q \)-subgroup is therefore normal, and \( G \) is a [semidirect product](https://www.edgepedia.org/semidirect-product) \( C_{q} \rtimes C_{p} \). If \( p \) does not divide \( q - 1 \), the action is trivial and \( G \) is cyclic of order \( pq \).<sup>[10](https://people.math.binghamton.edu/mazur/teach/50324/sylow.pdf)</sup>

**Nonexistence of simple groups.** A group of order \( 200 = 5^{2} \times 8 \) has \( n_{5} \equiv 1 \pmod{5} \) with \( n_{5} \) dividing 8, forcing \( n_{5} = 1 \); the unique Sylow 5-subgroup of order 25 is normal, so no such group is simple.<sup>[2](https://crypto.stanford.edu/pbc/notes/group/sylow.html)</sup> More broadly, Sylow arguments show that no group of non-prime order less than 60 is simple.<sup>[11](https://arxiv.org/abs/2311.08854)</sup>

## History

Ludvig Sylow (1832–1918), a Norwegian mathematician who spent most of his career as a schoolteacher in [Christiania](https://www.edgepedia.org/christiania) (now Oslo), had posed himself the underlying problem about subgroups of a given order during a course on [Galois theory](https://www.edgepedia.org/galois-theory) he gave in 1862–1863.<sup>[12](https://www.sciencedirect.com/science/article/pii/S0315086096900178)</sup> He had proved his theorem by September 1870 and communicated it privately to [Camille Jordan](https://www.edgepedia.org/camille-jordan) when Jordan visited Kristiania in 1872; Jordan, corresponding from a Swedish canal steamer, sent Sylow applications of the theorem and wrote to Felix Clebsch, editor of *Mathematische Annalen*, to hasten publication.<sup>[13](https://mathshistory.st-andrews.ac.uk/Extras/Holst_mathematics/)</sup> The paper, *Théorèmes sur les groupes de substitutions*, appeared in 1872 in volume 5 of the journal, pages 584–594.<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Sylow/)</sup>

Historical studies of the discovery record that Sylow built on a then almost unnoticed theorem of Cauchy and used methods from Galois theory in his proofs, in particular Galois' criterion for the solvability of equations of prime degree.<sup>[13](https://mathshistory.st-andrews.ac.uk/Extras/Holst_mathematics/)</sup><sup> • </sup><sup>[14](https://www.sciencedirect.com/science/article/pii/0315086088900481)</sup> The memoir's appearance marks a turning point in the contemporary assessment of finite group theory: Burnside later wrote that no considerable progress in the theory, apart from its applications, was made until the appearance of Sylow's 1872 memoir.<sup>[15](https://mathshistory.st-andrews.ac.uk/Extras/Burnside_groups/)</sup> Sylow's fame today rests on this one 10-page paper; his other major work was the edition of Abel's complete works prepared with [Sophus Lie](https://www.edgepedia.org/sophus-lie) between 1873 and 1881, and he held a specially created university chair at Christiania only from 1898, at age 65, teaching there for about twenty years.<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Sylow/)</sup>

## References

1. [Lecture 22: The Sylow Theorems, RES.18-011 Algebra I Student Notes, MIT OpenCourseWare](https://ocw.mit.edu/courses/res-18-011-algebra-i-student-notes-fall-2021/mit18_701f21_full_lec_new.pdf)
2. [Group Theory – Sylow Groups, Stanford lecture notes](https://crypto.stanford.edu/pbc/notes/group/sylow.html)
3. [Ludwig Sylow (1832–1918), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Sylow/)
4. [Group Theory review notes, MTHM024, Peter J. Cameron, Queen Mary University of London](https://webspace.maths.qmul.ac.uk/p.j.cameron/MTHM024/gtrev.pdf)
5. [Abstract Algebra, Chapter 4: Advanced group theory, Romyar Sharifi, UCLA](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)
6. [200ab course notes, D. Rogalski, UC San Diego](https://mathweb.ucsd.edu/~drogalsk/200ab-coursenotes-full.pdf)
7. [Lecture 23: Proofs and Applications of the Sylow Theorems, MIT OpenCourseWare](https://ocw.mit.edu/courses/res-18-011-algebra-i-student-notes-fall-2021/mit18_701f21_lect23.pdf)
8. [More on the Sylow Theorems, Keith Conrad, University of Connecticut](https://kconrad.math.uconn.edu/blurbs/grouptheory/sylowmore.pdf)
9. [Sylow Theorems, Historical Note, ProofWiki](https://proofwiki.org/wiki/Sylow_Theorems/Historical_Note)
10. [Applications of Sylow Theorem, M. Mazur, Binghamton University](https://people.math.binghamton.edu/mazur/teach/50324/sylow.pdf)
11. [A Formalization of Finite Group Theory: Part III, arXiv](https://arxiv.org/abs/2311.08854)
12. [Ludvig Sylow's Lectures on Algebraic Equations and Substitutions, Christiania (Oslo), 1862, Historia Mathematica](https://www.sciencedirect.com/science/article/pii/S0315086096900178)
13. [Holst on mathematics, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Holst_mathematics/)
14. [Die Entdeckung der Sylow-Sätze, Historia Mathematica](https://www.sciencedirect.com/science/article/pii/0315086088900481)
15. [W. Burnside, Theory of Groups of Finite Order, MacTutor extract](https://mathshistory.st-andrews.ac.uk/Extras/Burnside_groups/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Finite groups and classification › Subgroup structure and Sylow theory*

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