Finite supersolvable group: Difference between revisions

Revision as of 01:09, 26 December 2015

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties

Definition

A finite supersolvable group is a finite group satisfying the following equivalent conditions:

It is a supersolvable group: it has a normal series where all the quotients are cyclic groups.
It has a chief series where all the successive quotients are groups of prime order.
It is a solvable group that also satisfies the property that its chief series are composition series.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite abelian group	finite and an abelian group: any two elements commute		symmetric group:S3 is a counterexample	\|FULL LIST, MORE INFO
finite nilpotent group	finite and a nilpotent group		symmetric group:S3 is a counterexample	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group having subgroups of all orders dividing the group order	for every natural number dividing the order, there is a subgroup with that natural number as order	finite supersolvable implies subgroups of all orders dividing the group order	subgroups of all orders dividing the group order not implies supersolvable	\|FULL LIST, MORE INFO
finite solvable group	finite and a solvable group. This only requires a chief series with abelian quotients, or a composition series with cyclic quotients			\|FULL LIST, MORE INFO

@@ Line 13: / Line 13: @@
 ===Stronger properties===
-* [[Weaker than::Finite abelian group]]
+{| class="sortable" border="1"
-* [[Weaker than::Finite nilpotent group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::finite abelian group]] || finite and an [[abelian group]]: any two elements commute || || [[symmetric group:S3]] is a counterexample || {{intermediate notions short|finite supersolvable group|finite abelian group}}
+|-
+| [[Weaker than::finite nilpotent group]] || finite and a [[nilpotent group]] || || [[symmetric group:S3]] is a counterexample || {{intermediate notions short|finite supersolvable group|finite nilpotent group}}
+|}
 ===Weaker properties===
-* [[Stronger than::Group having subgroups of all orders dividing the group order]]: {{proofofstrictimplicationat|[[Finite supersolvable implies subgroups of all orders dividing the group order]]|[[Subgroups of all orders dividing the group order not implies supersolvable]]}}
+{| class="sortable" border="1"
-* [[Stronger than::Finite solvable group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::group having subgroups of all orders dividing the group order]] || for every natural number dividing the [[order of a group|order]], there is a subgroup with that natural number as order || [[finite supersolvable implies subgroups of all orders dividing the group order]]|| [[subgroups of all orders dividing the group order not implies supersolvable]] || {{intermediate notions short|group having subgroups of all orders dividing the group order|finite supersolvable group}}
+|-
+| [[Stronger than::finite solvable group]] || finite and a [[solvable group]]. This only requires a chief series with ''abelian'' quotients, or a ''composition'' series with cyclic quotients || || || {{intermediate notions short|finite solvable group|finite supersolvable group}}
+|}