One of the main types of algebraic systems (cf.
Algebraic system).
The theory of groups studies in the most
general form properties of algebraic operations which
are often encountered in mathematics and their
applications; examples of such operations are
multiplication of numbers, addition of vectors, successive
performance (composition) of transformations, etc. The concept of
a group is historically one of the first examples of abstract algebraic
systems and served, in many respects, as a
model for the restructuring of other mathematical disciplines at
the turn into the
20th century,
as a result
of which the concept of a mathematical system (a
structure) has become a fundamental concept in mathematics.
Definition.
A
group
is a non-empty set
with one binary operation that satisfies the following
axioms (the operation being written as multiplication):
1)
the operation is
associative,
i.e.
for any
,
and
in
;
2)
the operation admits a unit, i.e.
has an element
,
known as the
unit element,
such that
for any
in
;
3)
the operation admits inverse elements, i.e. for any
in
there exists an element
in
,
said to be
inverse
to
,
such that
.
The system of axioms 1)–3) is sometimes replaced
by an equivalent system of two axioms: 1); and 4)
the operation admits left and right quotients, i.e. for any two elements
,
in
there exist elements
,
in
,
the
left quotient
and the
right quotient
of division of
by
,
such that
,
.
It follows from this definition that the unit element in
any group is unique, that the element inverse to any
given element in the group is unique and that for any elements
,
of
both fractions obtained by dividing
by
are unique.
Historical remarks.
The origins of the idea of a group are
encountered in a number of disciplines, the principal one
being the theory of solving algebraic equations by radicals.
Permutations
were first employed to satisfy the needs of this theory by
J.L. Lagrange
(1771)
in his
Memoir on the algebraic solution of equations,
and in a paper by
A. Vandermonde
(1771).
It is the former paper
which is of special importance in group theory, since it gives, in terms of
polynomials,
what is really a decomposition of a symmetric
permutation group
into (right) cosets with respect to subgroups. The deep connections
between the properties of permutation groups and those of
equations were pointed out by
N.H. Abel
(1824)
and
by
E. Galois
(1830).
Galois must be credited with
concrete advances in group theory: the discovery of
the role played by normal subgroups (cf.
Normal subgroup)
in problems of solvability of equations by radicals,
the discovery that the alternating groups (cf.
Alternating group)
of order
are simple, etc.
C. Jordan's
treatise
(1870)
on
permutation groups played an important role in the
systematization and development of this branch of algebra.
The idea of a group arose in
geometry,
in an independent manner, when the only then existing antique geometry
had been replaced in the middle of the
19th century
by numerous other
"geometries" ,
and finding relations
between them had become an urgent problem. This question was solved
by studies in projective geometry, which dealt with the behaviour
of geometric figures under various transformations. The stress
in these studies gradually shifted to the
study of the transformations themselves and their
classification. Such a
"study of geometric mappings"
was extensively conducted by
A. Möbius,
who investigated
congruence,
similarity,
affinity,
collineation,
and, finally,
"elementary types of mappings"
of geometric figures,
that is, actually, their topological equivalence.
A.L. Cayley
(1854
and later) and other representatives of the English
school of the theory of invariants (cf.
Invariants, theory of)
gave a more systematic classification of geometries: Cayley
explicitly used the term
"group" ,
made systematic use
of the multiplication table which now carries his name (cf.
Cayley table),
proved that any
finite group
can be represented by permutations, and conceived a group as a system
which is defined by its generating elements and defining
relations. The final stage in this development was the
Erlangen program
of
F. Klein
(1872),
who based the classification of geometries on the concept of a
transformation group.
Number theory
is the third source of the concept of a group. As early as
1761
L. Euler,
in his study of
residues remaining in power division,
actually used congruences (cf.
Congruence)
and their division into
residue classes,
which in group-theoretic language means the decomposition of groups
into cosets of subgroups.
C.F. Gauss,
in his
Disquisitiones arithmeticae,
studied the cyclotomic equations (cf.
Cyclotomic polynomials)
and in fact determined subgroups of their Galois groups (cf.
Galois group).
He also studied the
"composition of binary quadratic forms"
in this context, and showed, in essence, that
the classes of equivalent forms form a finite
Abelian group
with respect to composition.
Towards the end of the
19th century
it was
recognized that the group-theoretic ideas employed for a long
time in various fields of mathematics were essentially the
same, and the modern abstract idea of the concept
of a group was finally developed. Thus, as early as
1895,
S. Lie
defined a group as a set of transformations
that is closed under an operation that is associative, admits
a unit element and inverse elements. The study of groups without assuming
them to be finite and without making any assumptions
as to the nature of their elements was
first formulated as an independent branch of mathematics with the appearance of the book
Abstract group theory
by
O.Yu. Shmidt
(1916).
Examples of groups.
The examples below illustrate the role played by groups in
algebra,
in other branches of mathematics and in natural sciences.
a)
Galois groups.
Let
be a finite, separable and normal
extension of a field
.
The automorphisms of
leaving the elements of
fixed form a group
with respect to composition, called the
Galois group
of the extension
.
The principal theorem in
Galois theory
states that the mapping which associates to every subgroup of
its fixed subfield (i.e. the subfield of
whose elements are fixed under the subgroup of
)
is an anti-isomorphism of the lattice of subgroups of
onto the lattice of intermediate subfields between
and
.
The application to the problem on the solvability
of equations by radicals is as follows. Let
be a polynomial in
over
and let
be a splitting field (cf.
Splitting field of a polynomial)
of
.
The group
is called the Galois group of
over
(its elements are naturally formed by the permutations of the roots of the equation
).
The result is that the equation
is solvable in radicals if and only if the Galois group of
is solvable (cf.
Solvable group).
In this and other similar examples groups appear in the form of
automorphism groups
(cf.
Automorphism)
of mathematical structures. This is one of the
most important ways of appearance, ensuring groups a
special place in algebra. In the words of Galois,
automorphisms of arbitrary structures can always be
"grouped" ,
while
a ring structure or any other useful structure
on a set of automorphisms is successfully introduced in special cases only.
b)
Homology groups.
The leading idea in homology theory is the application of the
theory of (Abelian) groups to the study of a category of
topological spaces.
To each space
is associated a family of Abelian groups
while each continuous mapping
defines a family of
homomorphisms
,
.
The study of the homology groups
(cf.
Homology group)
and their homomorphisms by the tools of group theory
often makes it possible to deal with a topological problem. A typical
example is the extension problem: Is it possible to extend a mapping
,
defined on a subspace
of
,
to all of
,
i.e. is it possible to represent
as the composite of the imbedding
and some continuous mapping
?
If so, then in the homology spaces one has
,
i.e. each homomorphism
can be factored through
with a given homomorphism
.
If this algebraic problem is unsolvable, then the
initial topological problem is unsolvable as well. Important
positive results can be obtained in this way.
Homology groups illustrate another typical manner of
application of groups: the study of non-algebraic objects
by means of algebraic systems which reflect their
behaviour. This is in fact the fundamental method of
algebraic topology.
A similar method, in particular homology groups, is also used with
success in the study of algebraic systems themselves — groups,
rings, etc. (e.g. in the theory of group extensions).
c)
Symmetry groups.
The concept of a group makes it possible to
describe the symmetries of a given geometrical figure. To any
figure one associates the set of spatial transformations that
map it onto itself. This set is a group
under composition. It also characterizes the symmetry of the figure.
This was in fact the approach of
E.S. Fedorov
(1890)
to the problem of classification of regular spatial
systems of points, which is one of the basic problems in crystallography (cf.
Crystallography, mathematical).
There are only 17 plane crystallographic groups (cf.
Crystallographic group),
which were found directly; there are 230
-dimensional crystallographic groups, which could be exhaustively classified only
by the use of group theory. This is historically the first
example of the application of group theory to natural sciences.
Group theory plays a similar role in physics. Thus,
the state of a physical system is represented in
quantum mechanics
by a point in an infinite-dimensional vector space.
If the physical system passes from one
state into another, its representative point undergoes some
linear transformation.
The ideas of
symmetry
and the theory of group representations (cf.
Representation of a group)
are of prime importance here.
These examples illustrate the contribution of group theory
to all classifications where symmetry plays a role.
The study of symmetry is actually equivalent to
the study of automorphisms of (not necessarily
mathematical) systems, and for this reason group
theory is indispensable in solving such problems.
Important classes of groups.
The
"final objective"
of group theory is to describe
all group operations or, in other words, all
groups, up to isomorphism. Group theory comprises several
parts, which are often distinguished by special conditions
imposed on the group operation or by the
introduction of additional structures into the group, related
in some way with the group operation.
The oldest branch of group theory, which is
still intensively studied, is the theory of finite groups (cf.
Finite group).
One of its important tasks is to determine the finite simple groups (cf.
Simple finite group).
These include many classical groups of matrices over finite fields,
and also
"sporadic"
simple finite groups (Mathieu groups, cf.
Mathieu group,
etc.). At the other end there are finite solvable groups (cf.
Solvable group)
in which specific subgroup systems (Hall, Carter, etc., cf.
Carter subgroup;
Hall subgroup)
are usually studied, since these largely determine the
structure of the group itself. Finite groups often
appear as permutation groups or as matrix groups
over finite fields. A large independent branch of the
theory of finite groups is the study of representations by matrices and permutations.
A typical method of study of
infinite groups
is to impose on them some finiteness condition (cf.
Group with a finiteness condition).
Here, the main interest is centred on periodic groups,
locally finite groups,
groups with the maximum condition for subgroups (Noetherian groups),
groups with the minimum condition for
subgroups (Artinian groups), residually-finite groups, groups of finite rank (cf.
Rank of a group),
and finitely-generated groups (cf.
Periodic group;
Noetherian group;
Artinian group;
Residually-finite group;
Finitely-generated group).
In the study of Abelian groups (cf.
Abelian group)
important roles are played by complete Abelian groups,
torsion-free Abelian groups and periodic Abelian groups, and
inside these groups by pure subgroups and primary subgroups.
The study of any given Abelian group is reduced to a large extent to
the theories of the classes listed above with the aid of
the theory of extensions of Abelian groups, which
is mainly developed by homological methods (cf.
Extension of a group).
Broader than the class of Abelian groups are
the classes of nilpotent groups and of solvable groups (cf.
Nilpotent group;
Solvable group),
the theory of which has also reached a
fairly advanced stage. The most useful extensions of
nilpotency and solvability are local nilpotency (cf.
Locally nilpotent group),
local solvability (cf.
Locally solvable group)
and the
normalizer condition,
as well as numerous properties determined by the presence of subnormal systems (cf.
Subgroup system)
of various types in a group. Of importance are
special classes of solvable and nilpotent
groups: supersolvable groups, polycyclic groups (cf.
Supersolvable group;
Polycyclic group).
An important branch of group theory is the theory of
transformation groups,
including permutation groups and the theory of linear groups (cf.
Permutation group;
Linear group).
A number of important classes of groups is
defined by the introduction of additional structures
compatible with the group operation; this includes topological
groups, Lie groups, algebraic groups, and ordered groups (cf.
Topological group;
Lie group;
Algebraic group;
Ordered group).
Of the other classes of groups, the following are worthy
of mention: groups which are free in some variety (cf.
Free group),
complete groups (cf.
Complete group),
groups having some property residually (cf.
Residually-finite group),
groups defined by imposing conditions on their
generating elements and defining relations, and
groups distinguished by imposing certain conditions on the
lattice of subgroups.