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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915401.png" /> be a [[Finite group|finite group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915402.png" /> a subset of the prime numbers that divide the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915403.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915404.png" />. A Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915406.png" />-basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915407.png" /> is a collection of Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915408.png" />-subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s0915409.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154010.png" /> (cf. [[Sylow subgroup|Sylow subgroup]]), one for each prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154012.png" />, such that: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154013.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154014.png" />, then the order of every element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154015.png" /> (the subgroup generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154016.png" />) is a product of non-negative powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154018.png" /> is the set of all primes dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154019.png" />, one speaks of a complete Sylow basis. Two Sylow bases are conjugate if there is a single element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091540/s09154020.png" /> that by [[conjugation]] transforms all the groups of the first into those of the second. Hall's second theorem, [[#References|[a2]]], says that every finite solvable group has a complete Sylow basis, and that all these bases are conjugate. Conversely, if a finite group has a complete Sylow basis, then it is solvable (cf. also [[Solvable group|Solvable group]]).
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Let  $  G $
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be a [[Finite group|finite group]] and $  \pi $
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a subset of the prime numbers that divide the order $  n $
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of $  G $.  
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A Sylow $  \pi $-
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basis $  S $
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is a collection of Sylow $  p $-
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subgroups $  P _ {p} $
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of $  G $(
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cf. [[Sylow subgroup|Sylow subgroup]]), one for each prime $  p $
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in $  \pi $,  
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such that: If $  P _ {p _ {1}  } \dots P _ {p _ {r}  } $
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are in $  S $,  
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then the order of every element in $  \{ G _ {p _ {1}  } \dots G _ {p _ {r}  } \} $(
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the subgroup generated by $  P _ {p _ {1}  } \dots P _ {p _ {r}  } $)  
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is a product of non-negative powers of $  p _ {1} \dots p _ {r} $.  
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If $  \pi $
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is the set of all primes dividing $  n $,  
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one speaks of a complete Sylow basis. Two Sylow bases are conjugate if there is a single element of $  G $
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that by [[conjugation]] transforms all the groups of the first into those of the second. Hall's second theorem, [[#References|[a2]]], says that every finite solvable group has a complete Sylow basis, and that all these bases are conjugate. Conversely, if a finite group has a complete Sylow basis, then it is solvable (cf. also [[Solvable group|Solvable group]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''2''' , Chelsea  (1960)  pp. 195ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Hall,  "On the Sylow systems of a soluble group"  ''Proc. London Math. Soc.'' , '''43'''  (1937)  pp. 316–323</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''2''' , Chelsea  (1960)  pp. 195ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Hall,  "On the Sylow systems of a soluble group"  ''Proc. London Math. Soc.'' , '''43'''  (1937)  pp. 316–323</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


Let $ G $ be a finite group and $ \pi $ a subset of the prime numbers that divide the order $ n $ of $ G $. A Sylow $ \pi $- basis $ S $ is a collection of Sylow $ p $- subgroups $ P _ {p} $ of $ G $( cf. Sylow subgroup), one for each prime $ p $ in $ \pi $, such that: If $ P _ {p _ {1} } \dots P _ {p _ {r} } $ are in $ S $, then the order of every element in $ \{ G _ {p _ {1} } \dots G _ {p _ {r} } \} $( the subgroup generated by $ P _ {p _ {1} } \dots P _ {p _ {r} } $) is a product of non-negative powers of $ p _ {1} \dots p _ {r} $. If $ \pi $ is the set of all primes dividing $ n $, one speaks of a complete Sylow basis. Two Sylow bases are conjugate if there is a single element of $ G $ that by conjugation transforms all the groups of the first into those of the second. Hall's second theorem, [a2], says that every finite solvable group has a complete Sylow basis, and that all these bases are conjugate. Conversely, if a finite group has a complete Sylow basis, then it is solvable (cf. also Solvable group).

References

[a1] A.G. Kurosh, "The theory of groups" , 2 , Chelsea (1960) pp. 195ff (Translated from Russian)
[a2] P. Hall, "On the Sylow systems of a soluble group" Proc. London Math. Soc. , 43 (1937) pp. 316–323
How to Cite This Entry:
Sylow basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sylow_basis&oldid=48919