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 subgroup ...

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 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 ...

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 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''. A ...

order (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 s ...

) * ''G'' is a supersolvable group 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 grou ...

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 belong ...

), . 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 uni ...

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''& ...

, 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 de ...

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

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 groups 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, a verb form that has a subject, usually being inflected or marke ...

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 d ...

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 else i ...

, ''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 bina ...

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 ways ...

. 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

* * * * * * Properties of groups {{Abstract-algebra-stub