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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 <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]]).
  
 
====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>

Revision as of 21:18, 29 November 2014

Let be a finite group and a subset of the prime numbers that divide the order of . A Sylow -basis is a collection of Sylow -subgroups of (cf. Sylow subgroup), one for each prime in , such that: If are in , then the order of every element in (the subgroup generated by ) is a product of non-negative powers of . If is the set of all primes dividing , one speaks of a complete Sylow basis. Two Sylow bases are conjugate if there is a single element of 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=35139