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The historical origin of group theory goes back to the works of Évariste Galois ( 1830 ), concerning the problem of when an Algebraic Equation is soluble by Radical s. Previous to this work, groups were mainly studied concretely, in the form of Permutation s; some aspects of Abelian Group theory were known in the theory of Quadratic Form s.

Many of the objects investigated in mathematics turn out to be groups. These include familiar number systems, such as the Integer s, the Rational Number s, the Real Number s, and the Complex Number s under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Another important example is given by non-singular Matrices under multiplication, and more generally, Invertible Functions under Composition . Group theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems in its own right.

Groups underlie many other Algebraic Structure s such as Field s and Vector Space s. They are also important tools for studying Symmetry in all its forms; the principle that ''the symmetries of any object form a group'' is foundational for much mathematics. For these reasons, group theory is an important area in modern mathematics, and also one with many applications to Mathematical Physics (for example, in Particle Physics ).


HISTORY


See Group Theory .


BASIC DEFINITIONS


  • ) is a s below. "a --- b" represents the result of applying the operation --- to the ordered pair (''a'', ''b'') of elements of ''G''. The group axioms are the following:

  • '' a, b and c in G, (a --- b) --- c = a --- (b --- c).

  • '' an element e in G such that for all a in G, e --- a = a --- e = ''a''.

  • '' Inverse Element '': For all a in G, there is an element b in G such that a --- b = b --- a = ''e'', where e is the neutral element from the previous axiom.


You will often also see the Axiom :
  • '' Closure '': For all a and b in G, a --- b belongs to G.

    • The way that the definition above is phrased, this axiom is not necessary, since binary operations are already required to satisfy closure.

  • is a group operation, however, it is nonetheless necessary to verify that --- satisfies closure; this is part of verifying that it is in fact a binary operation.


The neutral element is usually called the Identity Element for a multiplicative group and the Null Element or Zero Element for an additive group.

It can be shown that if there is both a left and a right neutral element, they must be equal (here both e) and there can be only one. This is why we can refer to the neutral element in the third axiom of the definition of a group, even though the second axiom only postulates that the set of the neutral elements is nonempty.

  •  ''e''  =  ''a'') and right inverse (''a'' --- ''b''  =  ''e'') imply the left identity and left inverse as given above. It is canonical to define the axioms as above because combinations of the above define other useful algebraic structures -- e.g., the Groupoid and Semigroup . Thus the above axioms are not strictly minimal from a logical viewpoint; however, the difference is slight and in practice one usually just checks the above axioms.


  • b ≠ b --- a. A group ''G'' is said to be '' Abelian '' (after the mathematician Niels Abel ) (or ''commutative'') if for every ''a'', ''b'' in ''G'', a --- b = b --- a. Groups lacking this property are called ''non-abelian''.