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Three theorems on maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915601.png" />-subgroups in a finite group, proved by L. Sylow [[#References|[1]]] and playing a major role in the theory of finite groups. Sometimes the union of all three theorems is called Sylow's theorem.
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Three theorems on maximal $p$-subgroups in a finite group, proved by L. Sylow [[#References|[1]]] and playing a major role in the theory of finite groups. Sometimes the union of all three theorems is called Sylow's theorem.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915602.png" /> be a finite group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915603.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915604.png" /> is a prime number not dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915605.png" />. Then the following theorems hold.
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Let $G$ be a finite group of order $p^ms$, where $p$ is a prime number not dividing $s$. Then the following theorems hold.
  
Sylow's first theorem: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915606.png" /> contains subgroups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915607.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915608.png" />; moreover, each subgroup of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s0915609.png" /> is a normal subgroup in at least one subgroup of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156010.png" />. This theorem implies, in particular, the following important results: there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156011.png" /> a [[Sylow subgroup|Sylow subgroup]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156012.png" />; any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156013.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156014.png" /> is contained in some Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156015.png" />-subgroup of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156016.png" />; the index of a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156017.png" />-subgroup is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156018.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156019.png" /> is a group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156020.png" />, then any of its proper subgroups is contained in some maximal subgroup of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156021.png" /> and all maximal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156022.png" /> are normal.
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Sylow's first theorem: $G$ contains subgroups of order $p^i$ for all $i=1,\ldots,m$; moreover, each subgroup of order $p^{i-1}$ is a normal subgroup in at least one subgroup of order $p^i$. This theorem implies, in particular, the following important results: there is in $G$ a [[Sylow subgroup|Sylow subgroup]] of order $p^m$; any $p$-subgroup of $G$ is contained in some Sylow $p$-subgroup of order $p^m$; the index of a Sylow $p$-subgroup is not divisible by $p$; if $G=P$ is a group of order $p^m$, then any of its proper subgroups is contained in some maximal subgroup of order $p^{m-1}$ and all maximal subgroups of $P$ are normal.
  
Sylow's second theorem: All Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156023.png" />-subgroups of a finite group are conjugate.
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Sylow's second theorem: All Sylow $p$-subgroups of a finite group are conjugate.
  
 
For infinite groups the analogous result is, in general, false.
 
For infinite groups the analogous result is, in general, false.
  
Sylow's third theorem: The number of Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156024.png" />-subgroups of a finite group divides the order of the group and is congruent to one modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156025.png" />.
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Sylow's third theorem: The number of Sylow $p$-subgroups of a finite group divides the order of the group and is congruent to one modulo $p$.
  
For arbitrary sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156026.png" /> of prime numbers, analogous theorems have been obtained only for finite solvable groups (see [[Hall subgroup|Hall subgroup]]). For non-solvable groups the situation is different. For example, in the alternating group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156027.png" /> of degree 5, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156028.png" /> there is a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156029.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156030.png" /> of order 6 whose index is divisible by a number from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156031.png" />. In addition, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156032.png" /> there is a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156033.png" />-subgroup isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156034.png" /> and not conjugate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156035.png" />. The number of Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156036.png" />-subgroups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156037.png" /> does not divide the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091560/s09156038.png" />.
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For arbitrary sets $\pi$ of prime numbers, analogous theorems have been obtained only for finite solvable groups (see [[Hall subgroup|Hall subgroup]]). For non-solvable groups the situation is different. For example, in the alternating group $A_5$ of degree 5, for $\pi=\{2,3\}$ there is a Sylow $\pi$-subgroup $S$ of order 6 whose index is divisible by a number from $\pi$. In addition, in $A_5$ there is a Sylow $\pi$-subgroup isomorphic to $A_4$ and not conjugate with $S$. The number of Sylow $\pi$-subgroups in $A_5$ does not divide the order of $A_5$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Sylow,  "Théorèmes sur les groupes de substitutions"  ''Math. Ann.'' , '''5'''  (1872)  pp. 584–594</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,  "Group theory" , Macmillan  (1959)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L. Sylow,  "Théorèmes sur les groupes de substitutions"  ''Math. Ann.'' , '''5'''  (1872)  pp. 584–594 {{ZBL|04.0056.02}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,  "Group theory" , Macmillan  (1959)</TD></TR>
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</table>
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[[Category:Group theory and generalizations]]

Latest revision as of 19:17, 4 April 2023

Three theorems on maximal $p$-subgroups in a finite group, proved by L. Sylow [1] and playing a major role in the theory of finite groups. Sometimes the union of all three theorems is called Sylow's theorem.

Let $G$ be a finite group of order $p^ms$, where $p$ is a prime number not dividing $s$. Then the following theorems hold.

Sylow's first theorem: $G$ contains subgroups of order $p^i$ for all $i=1,\ldots,m$; moreover, each subgroup of order $p^{i-1}$ is a normal subgroup in at least one subgroup of order $p^i$. This theorem implies, in particular, the following important results: there is in $G$ a Sylow subgroup of order $p^m$; any $p$-subgroup of $G$ is contained in some Sylow $p$-subgroup of order $p^m$; the index of a Sylow $p$-subgroup is not divisible by $p$; if $G=P$ is a group of order $p^m$, then any of its proper subgroups is contained in some maximal subgroup of order $p^{m-1}$ and all maximal subgroups of $P$ are normal.

Sylow's second theorem: All Sylow $p$-subgroups of a finite group are conjugate.

For infinite groups the analogous result is, in general, false.

Sylow's third theorem: The number of Sylow $p$-subgroups of a finite group divides the order of the group and is congruent to one modulo $p$.

For arbitrary sets $\pi$ of prime numbers, analogous theorems have been obtained only for finite solvable groups (see Hall subgroup). For non-solvable groups the situation is different. For example, in the alternating group $A_5$ of degree 5, for $\pi=\{2,3\}$ there is a Sylow $\pi$-subgroup $S$ of order 6 whose index is divisible by a number from $\pi$. In addition, in $A_5$ there is a Sylow $\pi$-subgroup isomorphic to $A_4$ and not conjugate with $S$. The number of Sylow $\pi$-subgroups in $A_5$ does not divide the order of $A_5$.

References

[1] L. Sylow, "Théorèmes sur les groupes de substitutions" Math. Ann. , 5 (1872) pp. 584–594 Zbl 04.0056.02
[2] M. Hall, "Group theory" , Macmillan (1959)
How to Cite This Entry:
Sylow theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sylow_theorems&oldid=11963
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article