Finite supersolvable group: Difference between revisions

Latest revision as of 04:21, 16 April 2017

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.
It is finite and a J-group, i.e., its lattice of subgroups satisfies the Jordan-Dedekind chain condition: all chains between two subgroups (with one contained in the other) have equal length.

Examples

Extreme examples

The trivial group is a finite supersolvable group.

Examples based on order

Any group of prime power order is a finite nilpotent group and hence a finite supersolvable group.
Any group whose order is square-free number is a metacyclic group and hence a finite supersolvable group.

Other examples

Any finite abelian group, and more generally, any finite nilpotent group, is a finite supersolvable group. Examples include any group of prime power order.
Any metacyclic group is supersolvable, and therefore, any finite metacyclic group is a finite supersolvable group. Examples include all finite dihedral groups, dicyclic groups, as well as general affine group of degree one over a finite field when the field is a finite prime field (i.e., any group of the form $GA(1,p)$ for a prime number $p$ ).

Non-examples

The smallest example of a finite non-supersolvable group is alternating group:A4. In particular, it has no nontrivial cyclic normal subgroup, and therefore cannot be supersolvable. Therefore, the groups symmetric group:S4 and special linear group:SL(2,3) (which have $A_{4}$ as a subgroup and quotient respectively) are also non-supersolvable.
In general, any finite non-solvable group is non-supersolvable. In particular, any group that contains a finite simple non-abelian group as a subgroup, quotient group, or subquotient is not supersolvable. Examples include the alternating group:A5 (the smallest finite simple non-abelian group), symmetric group:S5, and special linear group:SL(2,5)

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes		If $G$ is a finite supersolvable group and $H$ is a subgroup of $G$ , then $H$ is also a finite supersolvable group.
quotient-closed group property	Yes		If $G$ is a finite supersolvable group and $H$ is a normal subgroup of $G$ , then the quotient group $G/H$ is also a finite supersolvable group.
finite direct product-closed group property	Yes		If $G_{1},G_{2},\dots ,G_{n}$ are all finite supersolvable groups, the external direct product $G_{1}\times G_{2}\times \dots \times G_{n}$ is also a finite supersolvable group.
lattice-determined group property	Yes	follows from characterization as finite J-group	If $G_{1},G_{2}$ have isomorphic lattices of subgroups, then either both are finite supersolvable, or neither is.

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 8: / Line 8: @@
 # It has a [[defining ingredient::chief series]] where all the successive quotients are [[group of prime order|groups of prime order]].
 # It is a [[defining ingredient::solvable group]] that also satisfies the property that [[group whose chief series are composition series|its chief series are composition series]].
+# It is finite and a [[defining ingredient::J-group]], i.e., its [[lattice of subgroups]] satisfies the Jordan-Dedekind chain condition: all chains between two subgroups (with one contained in the other) have equal length.
+==Examples==
+===Extreme examples===
+* The [[trivial group]] is a finite supersolvable group.
+===Examples based on order===
+* Any [[group of prime power order]] is a [[finite nilpotent group]] and hence a finite supersolvable group.
+* Any group whose order is square-free number is a metacyclic group and hence a finite supersolvable group.
+===Other examples===
+* Any [[finite abelian group]], and more generally, any [[finite nilpotent group]], is a finite supersolvable group. Examples include any [[group of prime power order]].
+* Any [[metacyclic group]] is supersolvable, and therefore, any [[finite metacyclic group]] is a finite supersolvable group. Examples include all finite [[dihedral group]]s, [[dicyclic group]]s, as well as [[general affine group of degree one over a finite field|general affine group of degree one over a finite field]] when the field is a finite prime field (i.e., any group of the form <math>GA(1,p)</math> for a [[prime number]] <math>p</math>).
+===Non-examples===
+* The smallest example of a finite non-supersolvable group is [[alternating group:A4]]. In particular, it has no nontrivial [[cyclic normal subgroup]], and therefore cannot be supersolvable. Therefore, the groups [[symmetric group:S4]] and [[special linear group:SL(2,3)]] (which have <math>A_4</math> as a subgroup and quotient respectively) are also non-supersolvable.
+* In general, any finite non-solvable group is non-supersolvable. In particular, any group that contains a finite [[simple non-abelian group]] as a [[subgroup]], [[quotient group]], or [[subquotient]] is not supersolvable. Examples include the [[alternating group:A5]] (the smallest finite simple non-abelian group), [[symmetric group:S5]], and [[special linear group:SL(2,5)]]
+==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::subgroup-closed group property]] || Yes || || If <math>G</math> is a finite supersolvable group and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> is also a finite supersolvable group.
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || || If <math>G</math> is a finite supersolvable group and <math>H</math> is a normal subgroup of <math>G</math>, then the quotient group <math>G/H</math> is also a finite supersolvable group.
+|-
+| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || || If <math>G_1, G_2, \dots, G_n</math> are all finite supersolvable groups, the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math> is also a finite supersolvable group.
+|-
+| [[satisfies metaproperty::lattice-determined group property]] || Yes || follows from characterization as finite J-group || If <math>G_1, G_2</math> have isomorphic lattices of subgroups, then either both are finite supersolvable, or neither is.
+|}
 ==Relation with other properties==
 ===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}}
+|}