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''of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n0675803.png" />''
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A normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n0675804.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n0675805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n0675806.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n0675807.png" /> is a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n0675808.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n0675809.png" /> (see [[Sylow subgroup|Sylow subgroup]]). A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758010.png" /> has a normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758011.png" />-complement if some Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758012.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758014.png" /> lies in the centre of its normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]) (Burnside's theorem). A necessary and sufficient condition for the existence of a normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758015.png" />-complement in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758016.png" /> is given by Frobenius' theorem: A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758018.png" /> has a normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758019.png" />-complement if and only either for any non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758020.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758022.png" /> the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758023.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758024.png" />-group (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758025.png" /> is the normalizer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758026.png" /> the [[Centralizer|centralizer]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758028.png" />) or if for every non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758029.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758031.png" /> the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758032.png" /> has a normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758033.png" />-complement.
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 +
''of a finite group  $  G $''
 +
 
 +
A normal subgroup $  A $
 +
such that $  G = AS $
 +
and $  A \cap S = 1 $,  
 +
where $  S $
 +
is a Sylow $  p $-
 +
subgroup of $  G $(
 +
see [[Sylow subgroup|Sylow subgroup]]). A group $  G $
 +
has a normal $  p $-
 +
complement if some Sylow $  p $-
 +
subgroup $  S $
 +
of $  G $
 +
lies in the centre of its normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]) (Burnside's theorem). A necessary and sufficient condition for the existence of a normal $  p $-
 +
complement in a group $  G $
 +
is given by Frobenius' theorem: A group $  G $
 +
has a normal $  p $-
 +
complement if and only either for any non-trivial $  p $-
 +
subgroup $  H $
 +
of $  G $
 +
the quotient group $  N _ {G} ( H)/ C _ {G} ( H) $
 +
is a $  p $-
 +
group (where $  N _ {G} ( H) $
 +
is the normalizer and $  C _ {G} ( H) $
 +
the [[Centralizer|centralizer]] of $  H $
 +
in $  G $)  
 +
or if for every non-trivial $  p $-
 +
subgroup $  H $
 +
of $  G $
 +
the subgroup $  N _ {G} ( H) $
 +
has a normal $  p $-
 +
complement.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Harper &amp; Row  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Harper &amp; Row  (1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758034.png" /> be a group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758035.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758036.png" /> be the highest power of a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758037.png" /> dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758038.png" />. A subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758039.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758040.png" /> (and hence of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758041.png" />) is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758043.png" />-complement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758044.png" />. A normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758045.png" />-complement is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758046.png" />-complement that is normal. A finite group is solvable if and only if it has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758047.png" />-complement for every prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067580/n06758048.png" /> dividing its order. Cf. [[#References|[a1]]], [[#References|[a2]]] for more details; cf. also [[Hall subgroup|Hall subgroup]].
+
Let $  G $
 +
be a group of order n $
 +
and let $  p  ^ {e} $
 +
be the highest power of a prime number $  p $
 +
dividing n $.  
 +
A subgroup of $  G $
 +
of index $  p  ^ {e} $(
 +
and hence of order $  p  ^ {-} e n $)  
 +
is called a $  p $-
 +
complement in $  G $.  
 +
A normal $  p $-
 +
complement is a $  p $-
 +
complement that is normal. A finite group is solvable if and only if it has a $  p $-
 +
complement for every prime number $  p $
 +
dividing its order. Cf. [[#References|[a1]]], [[#References|[a2]]] for more details; cf. also [[Hall subgroup|Hall subgroup]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)  pp. Sect. 9.3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)  pp. Sect. VI.1</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)  pp. Sect. 9.3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)  pp. Sect. VI.1</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


of a finite group $ G $

A normal subgroup $ A $ such that $ G = AS $ and $ A \cap S = 1 $, where $ S $ is a Sylow $ p $- subgroup of $ G $( see Sylow subgroup). A group $ G $ has a normal $ p $- complement if some Sylow $ p $- subgroup $ S $ of $ G $ lies in the centre of its normalizer (cf. Normalizer of a subset) (Burnside's theorem). A necessary and sufficient condition for the existence of a normal $ p $- complement in a group $ G $ is given by Frobenius' theorem: A group $ G $ has a normal $ p $- complement if and only either for any non-trivial $ p $- subgroup $ H $ of $ G $ the quotient group $ N _ {G} ( H)/ C _ {G} ( H) $ is a $ p $- group (where $ N _ {G} ( H) $ is the normalizer and $ C _ {G} ( H) $ the centralizer of $ H $ in $ G $) or if for every non-trivial $ p $- subgroup $ H $ of $ G $ the subgroup $ N _ {G} ( H) $ has a normal $ p $- complement.

References

[1] D. Gorenstein, "Finite groups" , Harper & Row (1968)

Comments

Let $ G $ be a group of order $ n $ and let $ p ^ {e} $ be the highest power of a prime number $ p $ dividing $ n $. A subgroup of $ G $ of index $ p ^ {e} $( and hence of order $ p ^ {-} e n $) is called a $ p $- complement in $ G $. A normal $ p $- complement is a $ p $- complement that is normal. A finite group is solvable if and only if it has a $ p $- complement for every prime number $ p $ dividing its order. Cf. [a1], [a2] for more details; cf. also Hall subgroup.

References

[a1] M. Hall jr., "The theory of groups" , Macmillan (1959) pp. Sect. 9.3
[a2] B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. Sect. VI.1
How to Cite This Entry:
Normal p-complement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_p-complement&oldid=48016
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article