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# group

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1 See periodic table.

2 A mathematical structure consisting of a set of elements A, B, C, etc., for which there exists a law of composition, referred to as ‘multiplication’. Any two elements can be combined to give a ‘product’ AB. (1) Every product of two elements is an element of the set.(2) The operation is associative, i.e. A(BC) = (AB)C.(3) The set has an element I, called the identity element, such that IA = AI = A for all A in the set.(4) Each element of the set has an inverse A−1 belonging to the set such that AA−1 = A−1A = I.Although the law of combination is called ‘multiplication’ this does not necessarily have its usual meaning. For example, the set of integers forms a group if the law of composition is addition.

(1) Every product of two elements is an element of the set.

(2) The operation is associative, i.e. A(BC) = (AB)C.

(3) The set has an element I, called the identity element, such that IA = AI = A for all A in the set.

(4) Each element of the set has an inverse A−1 belonging to the set such that AA−1 = A−1A = I.

Two elements A, B of a group commute if AB = BA. If all the elements of a group commute with each other the group is said to be Abelian. If this is not the case the group is said to be non-Abelian.

The interest of group theory in physics and chemistry is in analysing symmetry. Discrete groups have a finite number of elements, such as the symmetries involved in rotations and reflections of molecules, which give rise to point groups. Continuous groups have an infinite number of elements where the elements are continuous. An example of a continuous group is the set of rotations about a fixed axis. The rotation group thus formed underlies the quantum theory of angular momentum, which has many applications to atoms and nuclei.

Subjects: Chemistry.

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