Solvable implies Fitting subgroup is self-centralizing - Revision history

Vipul at 06:18, 30 July 2013

2013-07-30T06:18:33Z

← Older revision		Revision as of 06:18, 30 July 2013
Line 4:		Line 4:
	property = self-centralizing subgroup}}		property = self-centralizing subgroup}}
	{{self-centralizing subgroup statement}}		{{self-centralizing subgroup statement}}
			[[Difficulty level::3\| ]]
	==Statement==		==Statement==

Vipul at 12:51, 14 October 2008

2008-10-14T12:51:08Z

← Older revision		Revision as of 12:51, 14 October 2008
Line 1:		Line 1:
	{{sdf subgroup property satisfaction in\|solvable group}}		{{sdf subgroup property satisfaction in\|
			group property = solvable group\|
			sdf = Fitting subgroup\|
			property = self-centralizing subgroup}}
	{{self-centralizing subgroup statement}}		{{self-centralizing subgroup statement}}
	==Statement==		==Statement==

Vipul at 15:24, 18 July 2008

2008-07-18T15:24:32Z

← Older revision		Revision as of 15:24, 18 July 2008
Line 1:		Line 1:
			{{sdf subgroup property satisfaction in\|solvable group}}
	{{self-centralizing subgroup statement}}		{{self-centralizing subgroup statement}}
	==Statement==		==Statement==

Vipul at 15:17, 14 July 2008

2008-07-14T15:17:01Z

← Older revision		Revision as of 15:17, 14 July 2008
Line 1:		Line 1:
			{{self-centralizing subgroup statement}}
	==Statement==		==Statement==

Vipul at 22:08, 13 July 2008

2008-07-13T22:08:19Z

← Older revision		Revision as of 22:08, 13 July 2008
Line 30:		Line 30:
	# Since <math>C</math> is already characteristic in <math>G</math>, and using the fact that characteristicity is transitive, we see that <math>B</math> is characteristic in <math>G</math>		# Since <math>C</math> is already characteristic in <math>G</math>, and using the fact that characteristicity is transitive, we see that <math>B</math> is characteristic in <math>G</math>

			Thus, <math>B</math> is a characteristic subgroup, hence a normal subgroup. So, <math>B</math> is a nilpotent normal subgroup. Moreover, <math>B \le C</math>, but <math>B</math> is not contained in <math>H</math>, so <math>B</math> cannot be contained in <math>F(G)</math>, contradicting the defining feature of <math>F(G)</math> as the subgroup generated by all nilpotent normal subgroups.
	==References==		==References==
	===Textbook references===		===Textbook references===
	* {{booklink-proved\|Gorenstein}}, Page 218, Theorem 1.3 (Section 6.1)		* {{booklink-proved\|Gorenstein}}, Page 218, Theorem 1.3 (Section 6.1)

Vipul: New page: ==Statement== In a solvable group, the Fitting subgroup is self-centralizing: it contains its centralizer in the whole group. ==Facts used== # [[uses::Characteristicity is c...

2008-07-13T22:06:56Z

New page: ==Statement== In a solvable group, the Fitting subgroup is self-centralizing: it contains its centralizer in the whole group. ==Facts used== # [[uses::Characteristicity is c...

New page

==Statement==

In a [[solvable group]], the [[Fitting subgroup]] is self-centralizing: it contains its [[centralizer]] in the whole group.

==Facts used==

# [[uses::Characteristicity is centralizer-closed]]
# [[uses::Characteristicity is intersection-closed]]
# Members of the derived series of a group are characteristic subgroups (follows from the fact that [[characteristicity is commutator-closed]])
# [[Characteristicity is quotient-transitive]]
# [[Characteristicity is transitive]]

==Proof==

'''Given''': A [[solvable group]] <math>G</math>. <math>F(G)</math> denotes the [[Fitting subgroup]] of <math>G</math>, and <math>C_G(F(G))</math> denotes its [[centralizer]] in <math>G</math>

'''To prove''': <math>C_G(F(G)) \le F(G)</math>

'''Proof''': Let <math>C = C_G(F(G))</math> and <math>H = C \cap F(G)</math>. Note that every element of <math>H</math> commutes with every element of <math>C</math>, so <math>H \le Z(C)</math>, and in particular, <math>H</math> is normal in <math>C</math>.

Consider the derived series of <math>C/H</math>. Since <math>G</math> is solvable, so is <math>C/H</math>, so its derived series terminates at the identity in finitely many steps. Let <math>B</math> be the inverse image of the term just before the trivial subgroup in this derived series. Then, <math>B \le G</math> is a subgroup with the property that <math>[B,B] \le H</math>. But since <math>H</math> commutes with every element of <math>C</math>, it also commutes with every element of <math>B</math>, so <math>[[B,B],B]</math> is trivial. Hence <math>B</math> is nilpotent of class two.

We now show that <math>B</math> is normal, through a series of observations:

# <math>F(G)</math> is a characteristic subgroup
# Since characteristicity is closed under taking centralizers, <math>C = C_G(F(G))</math> is also characteristic in <math>G</math>
# Since characteristicity is closed under intersections, <math>H</math> is characteristic in <math>G</math>
# The quotient <math>B/H</math> is a characteristic subgroup of <math>C/H</math>, being a member of the derived series
# Hence, using the fact that characteristicity is quotient-transitive, <math>B</math> is a characteristic subgroup of <math>C</math>
# Since <math>C</math> is already characteristic in <math>G</math>, and using the fact that characteristicity is transitive, we see that <math>B</math> is characteristic in <math>G</math>

==References==
===Textbook references===
* {{booklink-proved|Gorenstein}}, Page 218, Theorem 1.3 (Section 6.1)