Solvable Group
Considering the subgroup sequence
of group 
, and that 
is a normal subgroup of 
, if 
is a commutative group (Abelian group), then 
is solvable group.The symmetric group
is solvable group if 
and not solvable group if 
.Given a polynomial
, let 
be the decomposition of 
with fixed field 
.The Galois group
is an isomorphism of the symmetric group 
.Therefore, if
, the general polynomial 
cannot be algebraically solved.[0]Top