Finite solvable 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 group is termed a finite solvable group if it satisfies the following equivalent conditions:

1. It is a solvable group
2. It is a polycyclic group
3. It has Sylow complements for all prime divisors of the order of the group
4. It has Hall subgroups of all possible orders
5. All its composition factors (i.e., the quotient groups for any composition series for the group) are cyclic groups of prime order. Equivalently, all its composition factors are abelian.
6. All its chief factors (i.e., the successive quotient groups for any chief series for the group) are elementary abelian groups.

Equivalence of definitions

Further information: equivalence of definitions of finite solvable group

Examples

Extreme examples

• The trivial group is a finite solvable group.

Examples based on order

We call a natural number $n$ a solvability-forcing number if every group of order $n$ is solvable. It turns out that:

• Any prime power is solvability-forcing, because prime power order implies nilpotent and nilpotent implies solvable.
• Any product of two prime powers, i.e., any number of the form $p^{a}q^{b}$, with $p,q$ primes, is solvability-forcing. See order has only two prime factors implies solvable (this result is also termed Burnside's $p^{a}q^{b}$-theorem).
• Any odd number is solvability-forcing. See odd-order implies solvable. This result, also called the odd-order theorem or the Feit-Thompson theorem, is highly nontrivial.
• Any square-free number i.e., any number that is a product of pairwise distinct primes. See square-free implies solvability-forcing.

Non-examples

Any finite simple non-abelian group is a finite group that is not solvable. See classification of finite simple groups for a list of finite simple non-abelian groups.

Further, any group that contains a finite simple non-abelian group as a subgroup, has a finite simple non-abelian group as a quotient group, or admits a finite simple non-abelian group as a subquotient must be non-solvable.

The smallest order examples of finite non-solvable groups are below:

• alternating group:A5: This is the smallest order simple non-abelian group. It has order 60. See A5 is the simple non-abelian group of smallest order
• symmetric group:S5, special linear group:SL(2,5), direct product of A5 and Z2: All of these are groups of order 120 which have alternating group:A5 as one of their composition factors, and are hence neither simple nor solvable.
• projective special linear group:PSL(3,2): This is a simple non-abelian group of order 168.

Relation with other properties

Properties whose conjunction with finiteness gives this property

Below is a list of group properties such that a finite group has the property if and only if it is a finite solvable group.

Property	Meaning	Relation with solvability in general	Intermediate properties between it and solvability in general
solvable group	derived series reaches identity in finitely many steps	same	--
polycyclic group	has a subnormal series (of finite length) with cyclic quotient groups	stronger than solvability	|FULL LIST, MORE INFO
locally solvable group	every finitely generated subgroup is solvable	weaker than solvability	|FULL LIST, MORE INFO
hypoabelian group	the transfinite derived series reaches the identity element	weaker than solvability	|FULL LIST, MORE INFO
residually solvable group	every non-identity element is outside a normal subgroup for which the quotient group is solvable	weaker than solvability	|FULL LIST, MORE INFO

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite abelian group	finite and an abelian group	follows from abelian implies solvable	see solvable not implies abelian	|FULL LIST, MORE INFO
finite nilpotent group	finite and a nilpotent group	follows from nilpotent implies solvable	see solvable not implies nilpotent	|FULL LIST, MORE INFO
finite supersolvable group	finite and a supersolvable group	follows from supersolvable implies solvable	see solvable not implies supersolvable	|FULL LIST, MORE INFO
group having subgroups of all orders dividing the group order	finite and has subgroups of all orders dividing the group order	subgroups of all orders dividing the group order implies solvable		|FULL LIST, MORE INFO