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
A finite group is termed a finite solvable group if it satisfies the following equivalent conditions:
- It is a solvable group
- It is a polycyclic group
- It has Sylow complements for all prime divisors of the order of the group
- It has Hall subgroups of all possible orders
- 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.
- 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
- The trivial group is a finite solvable group.
Examples based on order
We call a natural number a solvability-forcing number if every group of order 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 , with primes, is solvability-forcing. See order has only two prime factors implies solvable (this result is also termed Burnside's -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
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
|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||Finitely generated solvable group, Finitely presented solvable group|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||Residually solvable group|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|
|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||Finite nilpotent group, Finite supersolvable group, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO|
|finite nilpotent group||finite and a nilpotent group||follows from nilpotent implies solvable||see solvable not implies nilpotent||Finite supersolvable group, Group having a Sylow tower, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO|
|finite supersolvable group||finite and a supersolvable group||follows from supersolvable implies solvable||see solvable not implies supersolvable||Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO|