Group theory describes the rules of operation of sets of numbers or objects, including finite, discrete, topological, and Lie groups. Group theory became one of the central tools in elementary particle physics in the 1960s. Although quantum electrodynamics (QED) operated as the fundamental theory of electricity and magnetism, physicists tried to adapt it to other interactions by making minor changes to it. Quantum electrodynamics (QED) was invariant under transformations that are contained in an Abelian group (i.e., a group on which the defined binary operation is cummutative). Physicists developed a theory similar to QED, except it is invariant under transformations in non-Abelian groups. Eventually, this theory was applied to the strong and weak interactions and incorporated into the standard model of particle physics.
A group is a set of objects, plus an operation that obeys a set of four mathematical rules. The objects may be numbers or something more complicated. The operation may be addition, multiplication, or something more abstract. For the sake of simplicity, the operation between two different elements A and B of a group will be written as AB. The first rule of a group is that if A and B are elements of a group, then AB is also an element of the group. The second rule states that the operation is associative, that is, (AB)C = A(BC). The third rule states that there is an identity element E of the group such that EA = AE = A. The fourth and final rule requires that in order to be a group, for any element A in the group, there is an inverse element A-1 such that AA-1 = A-1 A = E.
Groups are useful because the objects can be related to transformations of a system. For example, consider rotations in a plane. It can be shown that the set of rotations by any angle around a single axis in the plane form a group. First, if the plane is rotated by an angle (theta) and then rotated again by another angle (theta), the result is an element of the group corresponding to a single rotation by an angle ((theta) + (theta)). The second property, associativity, is clearly satisfied, because the angles add in rotations, and addition is associative. For the third rule, rotation by the angle 0 degrees is the identity element, because it leaves the system unchanged. The final rule is satisfied, because if a rotation is performed by angle (theta), another rotation can be performed by angle -(theta), to return to the original configuration.
A commutative group, as previously illustrated, simply means that the order of operations does not matter, that is, AB = BA. If the rotation group is enlarged, however, to include three-dimensional rotations, this is no longer the case; the order of rotations in different planes becomes important. Commutative groups are called Abelian while non-commutative groups are called non-Abelian.
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