Finite solvable group

Definition
A defining ingredient::finite group is termed a finite solvable group if it satisfies the following equivalent conditions:


 * 1) It is a defining ingredient::solvable group
 * 2) It is a defining ingredient::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 defining ingredient::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 defining ingredient::chief series for the group) are elementary abelian groups.

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 $$p^aq^b$$, with $$p,q$$ primes, is solvability-forcing. See order has only two prime factors implies solvable (this result is also termed Burnside's $$p^aq^b$$-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

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.

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.