Finite supersolvable group

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:

1. It is a supersolvable group: it has a normal series where all the quotients are cyclic groups.
2. It has a chief series where all the successive quotients are groups of prime order.
3. It is a solvable group that also satisfies the property that its chief series are composition series.
4. 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