Group theory

In mathematics and abstract algebra, **group theory** studies a type of algebraic structure called a **group**. Group theory is often used in mathematics as a starting point for the study of many algebraic structures, such as a set of numbers along with its addition and multiplication. Because group theory is also useful for studying symmetry in nature and abstract systems, it has many applications in physics and chemistry.

## Definition

A group is a set (collection) _G_ whose members are called _elements_. The elements can be numbers of some kind, or other abstract objects. The elements can even be material objects. There is also a binary operation which combines any two elements of _G_ and results in another element of _G_. This new element may be different from either of the original two, but it does not have to be: it only has to be an element of _G_. To be a group, the following four conditions must be true of _G_ and the operation defined over _G_:

- _Closure_: It is required to check if the suggested operation is actually an operation on the set. If an operation is used on any elements in the group, the element that is formed will also be part of the group.
    - For all _a_, _b_ in _G_, the result of the operation _a_ • _b_ is also in _G_.
- _Identity element_: One element of the group is special. It is called the identity element, often written as _e_. If the operation is used with the identity element and another element, the other element will not change.
    - There exists an element _e_ in _G_, such that for all elements _a_ in _G_, the equation _e_ • _a_ = _a_ • _e_ = _a_ holds.
- _Associativity_: When there are many operations, it does not matter in what order they are done, the result will be the same.
    - For all _a_, _b_ and _c_ in _G_, the equation (_a_ • _b_) • _c_ = _a_ • (_b_ • _c_) holds.
- _Inverse element_: Every element in the group has another element in the group known as its inverse. When the operation is performed between them, the result is the identity element.
    - For each _a_ in _G_, there exists an element _b_ in _G_ such that _a_ • _b_ = _b_ • _a_ = _e_, where _e_ is the identity element.

An important property about groups is whether the order of the elements matters when two elements are combined using the operation. Switching the order of two combined elements typically gives a different result. This means that we have many cases with _a_ • _b_ ≠ _b_ • _a_. However, if the order of a and b actually does not matter (for _any_ a and b, we _always_ have _a_ • _b_ = _b_ • _a_), then the group is called an abelian group (or commutative group).

If a group contains finitely many elements, then it is called a finite group. If a subset of a group is also a group, then the subset is called a subgroup.

## Examples

A simple example of a group is the set of integers (called **Z**) with the addition operator +. This group is called _G_ = (**Z**, +). Each of the required conditions is true of _G_, so _G_ is a group. If we replace • with the + sign, then we can easily check that these conditions are true:
- Closure: For any two integers _a_ and _b_ in _G_, _a_ + _b_ is also in _G_. For example, 3 and 5 are integers; and if we add them together, we see that 8 is an integer too.
- Identity element: For any integer _a_ in _G_, _a_ + 0 = _a_. So 0 is the identity element of G. For example, 69+0 is still 69.
- Associativity: For any three integers _a_, _b_, and _c_ in _G_, (_a_ + _b_) + _c_ = _a_ + (_b_ + _c_). For example, (1+3)+7 is the same as 1+(3+7).
- Inverse element: For any integer _a_ in _G_, there is another integer _b_ = −_a_ in _G_ such that _a_ + _b_ = 0. For example, 4 + (−4) = 0. In other words, −4 is the _inverse element_ of 4.

The addition operator is also commutative: the order of _a_ and _b_ does not matter. For any two integers _a_ and _b_ in G, _a_ + _b_ = _b_ + _a_. For example, 7+3 is the same as 3+7. Thus, _G_ is also an abelian group.

Commutative operators are common in our daily lives. For example, 7 · 3 is also the same as 3 · 7. However, not all operations are commutative. For example, 7÷3 is different from 3÷7 (and neither of these results is an integer), and 7<sup>3</sup> is different from 3<sup>7</sup>. Therefore, not all combinations of number sets and mathematical operations can form an abelian group—or even a group at all.

## Related pages

- Ring theory

## References

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