Everything about Group Mathematics totally explained
A
group is one of the fundamental objects of study in the field of
mathematics known as
abstract algebra, and more specifically
group theory.
Many structures investigated in mathematics turn out to be groups, including familiar number systems such as the
integers, or the
rational numbers with addition as the group operation, as well as the non-zero rational numbers with multiplication. Other important examples are groups of
matrices and
permutation groups. Almost all
algebraic structures such as
rings and
vector spaces can be defined concisely in terms of groups. Both relaxing and strengthening the requirements of the group axioms yields interesting further structures.
The
theory of groups allows for the properties of such structures to be investigated in a general and abstract setting. Beyond direct implications of the group axioms, basic techniques include studying groups related to a given one (such as
sub- or
quotient groups) or decomposing groups into
simpler parts. A particularly ample theory has been developed for
finite and for
abelian groups.
Symmetry groups of
geometrical objects, in particular
Lie groups, are also extensively used outside mathematics. Their ability to represent geometric transformations finds applications in
chemistry. Groups are essential abstractions in branches of
physics involving symmetry principles, such as
relativity,
quantum mechanics, and other fields.
Definition and illustration
A group (
G, •) is a
set G with a
binary operation • on
G that satisfies the following four
axioms:
» .
The abstract properties of Galois groups (in particular their
solvability) associated to polynomials give a criterion which polynomials do have all their solutions expressible by radicals.
Galois theory also explains the existence and structure of the formulae solving
cubic and
quartic equations.
Generalizations
In
abstract algebra, more general structures arise by relaxing some of the axioms defining a group:
Eliminating the requirement that every element have an inverse, then the resulting algebraic structure is called a monoid.
A monoid without an identity is called a semigroup.
Alternatively, relaxing the requirement that the operation be associative while still requiring the possibility of division, the resulting algebraic structure is a loop.
A loop without an identity is called a quasigroup.
Finally, dropping all axioms for the binary operation, the resulting algebraic structure is called a magma.
Groupoids, which are similar to groups except that the composition a • b need not be defined for all a and b, arise in the study of more involved kinds of symmetries, often in topological and analytical structures, for example the fundamental groupoid. Groupoids, in turn, are special sorts of categories.
Additionally:
Generalizing from binary to ternary operations leads to heaps.
The category of abelian groups form the prototype for the concept of an abelian category, which provides the general setting for homological algebra.
Groups may be viewed as a special case of a Hopf algebra, deforming a group in this larger context leads to the concept of a quantum group.
Formal group laws are certain formal power series which have properties much like a group operation.Further Information
Get more info on 'Group Mathematics'.
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