What is Group theory?
In [[mathematics]] and [[abstract algebra]], '''group theory''' studies a type of [[algebraic structure]] called a '''[[group (mathematics)|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''.<ref>{{Cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-10-12|website=Math Vault|language=en-US}}</ref> 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 {{nowrap begin}}''e'' • ''a'' = ''a'' • ''e'' = ''a''{{nowrap end}} holds. *''Associativity'': When there are many operations, it does not [[matter