In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the realm of group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, the term complemented group is used in two distinct, but similar ways.

In , a complemented group is one in which every subgroup

Subgroup

In group theory, 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 of G if the restriction of * to H x H is a group operation on H...

 has a group-theoretic complement. Such groups

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 are called completely factorizable groups in the Russian literature, following and .

The following are equivalent for any finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

 G:

• G is complemented
• G is a subgroup of a direct product of groups
  Direct product of groups
  In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
  
   of square-free order (group theory)
  Order (group theory)
  In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
  
   (a special type of Z-group
  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 groups whose Sylow subgroups are all cyclic....
  
  )
• G is a supersolvable group
  Supersolvable group
  In mathematics, a group is supersolvable 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...
  
   with elementary abelian
  Elementary Abelian group
  In group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; in particular it is a p-group....
  
   Sylow subgroups (a special type of A-group
  A-group
  A-Group is the designation for a distinct culture that arose between the First and Second Cataracts of the Nile in Nubia betweenthe Egyptian 1st dynasty and the 3rd millennium BC.The A-Group settled on very poor land with scarce natural resources, yet...
  
  ), .

Later, in , a group is said to be complemented if the lattice of subgroups

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....

 is a complemented lattice

Complemented lattice

In the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0....

, 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 = { hk : h in H, k in K } form a subgroup. Such groups are also called K-groups in the Italian and lattice theoretic literature, such as . The Frattini subgroup

Frattini subgroup

In mathematics, the Frattini subgroup Φ of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ = e...

 of a K-group is trivial; if a group has a core-free maximal subgroup

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 strictly. In other words H is a maximal element of the partially...

 that is a K-group, then it itself is a K-group; hence subgroups of K-grous need not be K-groups, but quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

s and direct products of K-groups are K-groups, . In it is shown that every finite simple group is a complemented group. Note that in the classification of finite simple groups

Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

, 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

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

 of order p², where p is a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. 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.