Groups
Taking the ideas of elements, sets, identity, associativity, commutativity and inverse elements together, gives one of the most important structures in mathematics: a group.
A group consists of:
* A set S of elements,
* A (closed) binary operation (just one)
* An identity element exists,
* Every element has an inverse,
* The operation is associative.
If commutativity is included as well, then we get an Abelian group.
For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, -a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c).
The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.
The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is 1/4, which is not an integer.
The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.
There are some structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative.
All groups are monoids, and all monoids are semigroups.
Rings and fields — Groups just have one binary operation. Rings and fields have two.
Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence). For the integers (a + b) × c = a×c+ b×c and c × (a + b) = c×a + c×b, and × is said to be distributive over +.
A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an Abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as -a.
The integers are an example of a ring. The integers have additional properties which make it an integral domain.
A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a-1.
The rational numbers, real number and complex numbers are all examples of fields.
