abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part 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 ( ...

since it arose in the 19th century. One major area of study has been classification: the

classification of finite simple groups In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...

(those with no nontrivial

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

) was completed in 2004.

History

During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete

was achieved, meaning that all those simple groups from which all finite groups can be built are now known. During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of

general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

s over

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

s. Finite groups often occur when considering

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s, which may be viewed as dealing with "

continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant u ...

", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

. The properties of finite groups can thus play a role in subjects such as

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

and

chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

Examples

Permutation groups

The symmetric group S_''n'' on a

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...

of ''n'' symbols is the group whose elements are all the

permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...

of the ''n'' symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself. Since there are ''n''! (''n''

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

) possible permutations of a set of ''n'' symbols, it follows that the order (the number of elements) of the symmetric group S_''n'' is ''n''!.

Cyclic groups

A cyclic group Z_''n'' is a group all of whose elements are powers of a particular element ''a'' where , the identity. A typical realization of this group is as the complex roots of unity. Sending ''a'' to a primitive root of unity gives an isomorphism between the two. This can be done with any finite cyclic group.

Finite abelian groups

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). They are named after Niels Henrik Abel. An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

Groups of Lie type

A group of Lie type is a group closely related to the group ''G''(''k'') of rational points of a reductive linear algebraic group ''G'' with values in the field ''k''. Finite groups of Lie type give the bulk of nonabelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups. Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic, symmetric and alternating groups, with the projective special linear groups over prime finite fields, PSL(2, ''p'') being constructed by Évariste Galois in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan's theorem that the projective special linear group PSL(2, ''q'') is simple for ''q'' ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(''n'', ''q'') of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley realized that after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraic groups over an arbitrary field ''k'', leading to construction of what are now called ''Chevalley groups''. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (''Tits simplicity theorem''). Although it was known since 19th century that other finite simple groups exist (for example, Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the sporadic groups, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their ''geometry'' in the sense of Tits. The belief has now become a theorem – the

. Inspection of the list of finite simple groups shows that groups of Lie type over a

include all the finite simple groups other than the cyclic groups, the alternating groups, the Tits group, and the 26 sporadic simple groups.

Main theorems

Lagrange's theorem

For any finite group ''G'', the order (number of elements) of every

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

''H'' of ''G'' divides the order of ''G''. The theorem is named after

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaArthur Cayley, states that every group ''G'' is isomorphic to a

of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

acting on ''G''. This can be understood as an example of the

group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...

of ''G'' on the elements of ''G''.

Burnside's theorem

Burnside's theorem in

states that if ''G'' is a finite group of order ''p'q'', where ''p'' and ''q'' are

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...

s, and ''a'' and ''b'' are non-negative

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s, then ''G'' is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes.

Feit–Thompson theorem

The Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by

Classification of finite simple groups

The

is a theorem stating that every finite simple group belongs to one of the following families: * A cyclic group with prime order; * An

alternating group In mathematics, an alternating group is the Group (mathematics), 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 ...

of degree at least 5; * A simple group of Lie type; * One of the 26 sporadic simple groups; * The Tits group (sometimes considered as a 27th sporadic group). The finite simple groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the

s are the basic building blocks of the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of

integer factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...

is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution. The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein (d.1992), Lyons, and

Solomon Solomon (), also called Jedidiah, was the fourth monarch of the Kingdom of Israel (united monarchy), Kingdom of Israel and Judah, according to the Hebrew Bible. The successor of his father David, he is described as having been the penultimate ...

are gradually publishing a simplified and revised version of the proof.

Number of groups of a given order

Given a positive integer ''n'', it is not at all a routine matter to determine how many

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

types of groups of order ''n'' there are. Every group of prime order is cyclic, because Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. If ''n'' is the square of a prime, then there are exactly two possible isomorphism types of group of order ''n'', both of which are abelian. If ''n'' is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order ''n'', and the number grows very rapidly as the power increases. Depending on the prime factorization of ''n'', some restrictions may be placed on the structure of groups of order ''n'', as a consequence, for example, of results such as the Sylow theorems. For example, every group of order ''pq'' is cyclic when are primes with not divisible by ''q''. For a necessary and sufficient condition, see cyclic number. If ''n'' is squarefree, then any group of order ''n'' is solvable. Burnside's theorem, proved using group characters, states that every group of order ''n'' is solvable when ''n'' is divisible by fewer than three distinct primes, i.e. if , where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are non-negative integers. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order ''n'' is solvable when ''n'' is odd. For every positive integer ''n'', most groups of order ''n'' are solvable. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the

. For any positive integer ''n'' there are at most two simple groups of order ''n'', and there are infinitely many positive integers ''n'' for which there are two non-isomorphic simple groups of order ''n''.

Table of distinct groups of order ''n''

References

External links

* * * * Small groups o
GroupNames
*

for groups of small order {{Authority control Properties of groups