A solvable group is a group having a normal series such that each normal factor is Abelian.
The special case of a solvable finite group is a
group whose composition indices are all prime numbers.
Solvable groups are sometimes called "soluble groups," a turn of phrase
that is a source of possible amusement to chemists.
The term "solvable" derives from this type of group's relationship to Galois's theorem, namely that the symmetric
group
is unsolvable for
while it is solvable for ,
2, 3, and 4. As a result, the polynomial equations
of degree
are (in general) not solvable using finite additions, multiplications, divisions,
and root extractions.
A major building block for the classification of finite simple groups was the Feit-Thompson theorem, which proved that every
group of odd order is solvable. This proof took up an entire journal issue.
Every finite group of order , every Abelian group,
and every subgroup of a solvable group is solvable.
Betten (1996) has computed a table of solvable groups of order up to 242 (Besche
and Eick 1999).
is unsolvable for 
while it is solvable for 
,
2, 3, and 4. As a result, the polynomial equations
of degree 
are (in general) not solvable using finite additions, multiplications, divisions,
and root extractions.
, every Abelian group,
and every subgroup of a solvable group is solvable.
Betten (1996) has computed a table of solvable groups of order up to 242 (Besche
and Eick 1999).
