In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
is supersolvable (or supersoluble) if it has an invariant
normal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups:
:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G
where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...
where all the factors are
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s. Supersolvability is stronger than the notion of
solvability.
Definition
Let ''G'' be a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. ''G'' is supersolvable if there exists a
normal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups:
:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G
where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...
:
such that each
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
is cyclic and each
is normal in
.
By contrast, for a
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...
the definition requires each quotient to be
abelian. In another direction, a
polycyclic group
In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational ...
must have a
subnormal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups:
:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G
where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...
with each quotient cyclic, but there is no requirement that each
be normal in
. 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
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
on four points,
, is solvable but not supersolvable.
Basic Properties
Some facts about supersolvable groups:
* Supersolvable groups are always
polycyclic, and hence
solvable.
* Every
finitely generated nilpotent group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with .
Intui ...
is supersolvable.
* Every
metacyclic group
In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group ''G'' for which there is a short exact sequence
:1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\,
where ''H'' and ''K'' ar ...
is supersolvable.
* The
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
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 {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
A ...
order, and every group with cyclic Sylow subgroups (a
Z-group In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups:
* in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic.
* in the s ...
), 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 In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 .
In this sec ...
.
*Every
maximal subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.
In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' s ...
in a supersolvable group has prime
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
.
*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
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion.
In this lattice, the join of two subgroups is the subgroup generated by their uni ...
of a group, and is sometimes called the
Jordan–Dedekind chain condition
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
.
* By
Baum's theorem
eBaum's World is an entertainment website owned by Literally Media. The site was founded in 2001 and features comedy content such as memes, viral videos, images, and other forms of Internet culture. Content is primarily user submitted in excha ...
, every supersolvable finite group has a
DFT algorithm running in time ''O''(''n'' log ''n'').
References
*Schenkman, Eugene. Group Theory. Krieger, 1975.
*Schmidt, Roland. Subgroup Lattices of Groups. de Gruyter, 1994.
*Keith Conrad, SUBGROUP SERIES II, Section 4 , http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/subgpseries2.pdf
Solvable groups
{{Abstract-algebra-stub