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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,   "Finite groups" , Harper &amp; Row  (1968)</TD></TR></table>

====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. [[#References|[a1]]], [[#References|[a2]]] for more details; cf. also [[Hall subgroup|Hall subgroup]].

====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>