In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.
Let G be a group. G is supersolvable if there exists a normal series
{
1
}
=
H
0
◃
H
1
◃
⋯
◃
H
s
−
1
◃
H
s
=
G
such that each quotient group
H
i
+
1
/
H
i
is cyclic and each
H
i
is normal in
G
.
By contrast, for a solvable group the definition requires each quotient to be abelian. In another direction, a polycyclic group must have a normal series with each quotient cyclic, but there is no requirement that each
H
i
be normal in
G
. As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions. For a concrete example, the alternating group on four points,
A
4
, is solvable but not supersolvable.
Some facts about supersolvable groups:
Supersolvable groups are always polycyclic, and hence solvable.
Every finitely generated nilpotent group is supersolvable.
Every metacyclic group is supersolvable.
The commutator subgroup of a supersolvable group is nilpotent.
Subgroups and quotient groups of supersolvable groups are supersolvable.
A finite supersolvable group has an invariant normal series with each factor cyclic of prime order.
In fact, the primes can be chosen in a nice order: For every prime p, and for π the set of primes greater than p, a finite supersolvable group has a unique Hall π-subgroup. Such groups are sometimes called ordered Sylow tower groups.
Every group of square-free order, and every group with cyclic Sylow subgroups (a Z-group), is supersolvable.
Every irreducible complex representation of a finite supersolvable group is monomial, that is, induced from a linear character of a subgroup. In other words, every finite supersolvable group is a monomial group.
Every maximal subgroup in a supersolvable group has prime index.
A finite group is supersolvable if and only if every maximal subgroup has prime index.
A finite group is supersolvable if and only if every maximal chain of subgroups has the same length. This is important to those interested in the lattice of subgroups of a group, and is sometimes called the Jordan–Dedekind chain condition.
By Baum's theorem, every supersolvable finite group has a DFT algorithm running in time O(n log n).
