Vipul: Created page with "==Definition== Let $G$ be a finite group and $p$ be a prime number. A '''p-complement''' (sometimes called a $p$ -Sylow complement) in..."

2012-03-22T23:46:12Z

Created page with "==Definition== Let <math>G</math> be a finite group and <math>p</math> be a prime number. A '''p-complement''' (sometimes called a <math>p</math>-Sylow complement) in..."

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==Definition==

Let <math>G</math> be a [[finite group]] and <math>p</math> be a [[prime number]]. A '''p-complement''' (sometimes called a <math>p</math>-Sylow complement) in <math>G</math> can be defined in the following equiavlent ways:

* It is a [[subgroup]] of <math>G</math> whose [[order of a group|order]] is relatively prime to <math>p</math> and whose index is a power of <math>p</math>.
* It is a subgroup whose order is the largest divisor of the order of <math>G</math> that is relatively prime to <math>p</math>.
* It is a <math>p'</math>-[[Hall subgroup]], i.e., a Hall subgroup in <math>G</math> for the set of all primes excluding <math>p</math>.
* It is a [[permutable complements|permutable complement]] to any <math>p</math>-[[Sylow subgroup]] of <math>G</math>.

Note that <math>p</math>-complements need not exist. It is also possible for a group to have more than one conjugacy class of <math>p</math>-complements. In fact, [[Hall's theorem]] shows that if <math>p</math>-complements exist in a finite group for all primes <math>p</math>, then the group is a [[finite solvable group]].

==Particular cases==

* If <math>p</math> does not divide the order of <math>G</math>, then the whole group <matH>G</matH> itself is the unique <math>p</math>-complement.
* If <math>G</matH> is a [[finite p-group]], then the trivial subgroup is the unique <math>p</math>-complement.
* There is a unique <math>p</math>-complement if and only if it is a [[normal p-complement]].

P-complement - Revision history

Vipul: Created page with "==Definition== Let G be a finite group and p be a prime number. A '''p-complement''' (sometimes called a p-Sylow complement) in..."

Vipul: Created page with "==Definition== Let $G$ be a finite group and $p$ be a prime number. A '''p-complement''' (sometimes called a $p$ -Sylow complement) in..."