View source for Characteristic p-functor that gives a characteristic subgroup

{{characteristic p-functor property}}

==Definition==

Suppose <math>p</math> is a [[prime number]] and <math>G</math> is a [[finite group]] such that <math>W</math> is a [[conjugacy functor]] for <math>G</math> for the prime <math>p</math> arising from a [[characteristic p-functor]]. We say that <math>W</math> is a '''characteristic p-functor that gives a characteristic subgroup''' if it satisfies the following equivalent conditions:

# For every pair of <math>p</math>-[[Sylow subgroup]]s <math>P,Q</math> of <math>G</math>, <math>W(P) = W(Q)</math>.
# For every pair of <math>p</math>-[[Sylow subgroup]]s <math>P,Q</math> of <math>G</math>, <math>W(P)</math> is a [[normal subgroup]] of <math>Q</math>.
# Each of these:
#* <math>W</math> is a [[weakly closed conjugacy functor]] and there exists a <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math> such that <math>W(P) \le O_p(G)</math> where <math>O_p(G)</math> is the [[p-core]] of <math>G</math>.
#*  <math>W</math> is a [[weakly closed conjugacy functor]] and for every <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math>, <math>W(P) \le O_p(G)</math> where <math>O_p(G)</math> is the <math>p</math>-core of <math>G</math>.
# Each of these:
#* There exists a <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math> such that <math>W(P)</math> is a [[characteristic subgroup]] of <math>G</math>.
#* For every <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math>, <math>W(P)</math> is a [[characteristic subgroup]] of <math>G</math>.
# Each of these:
#* There exists a <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math> such that <math>W(P)</math> is a [[normal subgroup]] of <math>G</math>.
#* For every <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math>, <math>W(P)</math> is a [[normal subgroup]] of <math>G</math>.

===Equivalence of definitions===

{{further|[[equivalence of normality and characteristicity conditions for conjugacy functor]]}}