In
mathematics, in the realm of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the term complemented group is used in two distinct, but similar ways.
In , a complemented group is one in which every
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
has a group-theoretic
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
. Such
groups are called completely factorizable groups in the Russian literature, following and .
The following are equivalent for any
finite group ''G'':
* ''G'' is complemented
* ''G'' is a subgroup of a
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of groups 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''.
...
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
(a special type of
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 ...
)
* ''G'' is a
supersolvable group
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.
Definition
Let ''G'' be a group. ''G'' is ...
with
elementary abelian
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian gr ...
Sylow subgroup
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...
s (a special type of
A-group
The A-Group culture was an ancient culture that flourished between the First and Second Cataracts of the Nile in Nubia. It lasted from 3800 BC to 3100 BC.
Overview
In 1907, the Egyptologist George A. Reisner first discovered artifacts belongi ...
), .
Later, in , a group is said to be complemented if 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 un ...
is a
complemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b''&n ...
, that is, if for every subgroup ''H'' there is a subgroup ''K'' such that ''H'' ∩ ''K'' = 1 and ⟨''H'', ''K'' ⟩ is the whole group. Hall's definition required in addition that ''H'' and ''K'' permute, that is, that ''HK'' = form a subgroup. Such groups are also called K-groups in the Italian and
lattice theoretic literature, such as . The
Frattini subgroup
In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is defi ...
of a K-group is
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
; if a group has a
core-free In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group.
The normal core Definition
For a group ''G'', the nor ...
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' ...
that is a K-group, then it itself is a K-group; hence subgroups of K-groups need not be K-groups, but
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 exam ...
s and direct products of K-groups are K-groups, . In it is shown that every
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past partici ...
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The da ...
is a complemented group. Note that in the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or els ...
, ''K''-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups.
An example of a group that is not complemented (in either sense) is the
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 bi ...
of order ''p''
2, where ''p'' is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
. This group only has one nontrivial subgroup ''H'', the cyclic group of order ''p'', so there can be no other subgroup ''L'' to be the complement of ''H''.
References
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Properties of groups
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