Group theory

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Group theory is an advanced subject in mathematics. It is the study of mathematical objects called groups. It is very important in many areas of science, especially chemistry and physics.

[change] Groups

In mathematics, a group is a collection of abstract objects, called elements. Every group also has an action associated with it, called an operation, which can combine two elements of the group to always form another element. This element may be different than the original two, but it doesn't have to be. It only has to be one of the elements of the group. To be a group, these three ideas also need to be true:

1. There is one special element called the identity. When the operation combines any element in the group with the identity, the same element results unchanged. It does not matter whether the identity is first or second for this to happen.
2. For any element, there is exactly one other element, called its inverse, in the group. When the operation combines an element and its inverse, in any order, the result is the identity element. The inverse usually is a different element, but doesn't have to be: the identity is its own inverse.
3. When combining three elements with the operation, it does not matter whether the first two are combined, then the result combined with the third, or the last two combined, then the first combined with the result. This is also true when combining 4 or more elements.

An important idea about a group is that when two elements are combined with the operation, the order of the elements matters. Switching the order of two combined elements gives a different result, except in special cases.