abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, 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 ...

of a

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 ...

is cyclic. Moreover, for a

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 marked ...

cyclic group of order ''n'', every subgroup's order is a divisor of ''n'', and there is exactly one subgroup for each divisor. This result has been called the fundamental theorem of cyclic groups.

Finite cyclic groups

For every finite group ''G'' of order ''n'', the following statements are equivalent: * ''G'' is cyclic. * For every divisor ''d'' of ''n'', ''G'' has at most one subgroup of order ''d''. If either (and thus both) are true, it follows that there exists exactly one subgroup of order ''d'', for any divisor of ''n''. This statement is known by various names such as characterization by subgroups. (See also

for some characterization.) There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the

Klein group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. ...

is an example. However, the Klein group has more than one subgroup of order 2, so it does not meet the conditions of the characterization.

The infinite cyclic group

The infinite cyclic group is isomorphic to the additive subgroup Z of the integers. There is one subgroup ''d''Z for each integer ''d'' (consisting of the multiples of ''d''), and with the exception of the trivial group (generated by ''d'' = 0) every such subgroup is itself an infinite cyclic group. Because the infinite cyclic group is a

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...

on one generator (and the trivial group is a free group on no generators), this result can be seen as a special case of the

Nielsen–Schreier theorem In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier. Statement of the theorem A free group may be defined from a grou ...

that every subgroup of a free group is itself free.. The fundamental theorem for finite cyclic groups can be established from the same theorem for the infinite cyclic groups, by viewing each finite cyclic group as a

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 ...

of the infinite cyclic group.

Lattice of subgroups

In both the finite and the infinite case, 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 union, a ...

of a cyclic group is isomorphic to the dual of a

divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

lattice. In the finite case, the lattice of subgroups of a cyclic group of order ''n'' is isomorphic to the dual of the lattice of divisors of ''n'', with a subgroup of order ''n''/''d'' for each divisor ''d''. The subgroup of order ''n''/''d'' is a subgroup of the subgroup of order ''n''/''e'' if and only if ''e'' is a divisor of ''d''. The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by ''d'' is a subgroup of the subgroup generated by ''e'' if and only if ''e'' is a divisor of ''d''. Divisibility lattices are

distributive lattice In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set uni ...

s, and therefore so are the lattices of subgroups of cyclic groups. This provides another alternative characterization of the finite cyclic groups: they are exactly the finite groups whose lattices of subgroups are distributive. More generally, a

finitely generated group In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...

is cyclic if and only if its lattice of subgroups is distributive and an arbitrary group is locally cyclic if and only its lattice of subgroups is distributive.. The additive group of the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s provides an example of a group that is locally cyclic, and that has a distributive lattice of subgroups, but that is not itself cyclic.

References

{{reflist Theorems in group theory Articles containing proofs