Definition of a group

A group consists of a set (of symmetry operations, numbers, etc.) together with a rule by which any two elements of the set may be combined - which will be called, generically, multiplication - with the following four properties  

1 Closure The result of combining any two elements - the product of any two elements - is another element in the set. 

3 There exists a unit element, or identity, denoted E, such that E a = afor any element of the group. 

4 For every element a of the group, the group contains another element called the inverse, a, such that a - a = E. Note that as E E = E, the inverse of E is. E itself 

Problem 3-1. Verify that the set of covering operations of the water molecule is a group, with the definition of the product of two operations as the compound operation resulting from applying them in succession. 

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The carbon atom in CO2 has two groups of electrons. Recall from our definition of a group that a double bond counts as one group of four electrons. Although each double bond includes four electrons, all four are concentrated between the nuclei. Remember also that the VSEPR model applies to electron groups, not specifically to electron pairs (despite the name of the model). It is the number of ligands and lone pairs, not the number of shared eiectrons, that determines the steric number and hence the molecular shape of an inner atom. 

The preceding definition of a group seems rather abstract and exceedingly general and in order to bring things down to earth, we give in this section some concrete examples. 

Note the convention that in forming each product, the element on the side of the table is put on the left. In the definition of a group, it was not postulated that AB = BA. Groups such that AB = BA for all pairs of elements are called commutative or Abelian. 

Selection rules of the vibrations of molecules and crystals 44 2.7.3.1 Definition of a group, multiplication tables 44... 

Each term requires a unique identifier (id), a name, as well as at least one is a relationship to another term. Additional data are optional in the example, a namespace, its definition, and a synonym. The namespace modifier, for instance, allows the definition of a group in which the term is valid and can also be defined for other modifiers, like the is a relationship. 

If P and Q are elements of a group, then so, by the definition of a group, is the inverse of Q, and consequently Q XPQ. If this latter combination is identical with the element R we can write... 

If a group G is has a finite number of elements, G is called a finite group. If it has an infinite number of elements, it is called an infinite group. Examples of finite groups include Zn, 5. and >2n- Ench one will be examined in detail to show how they satisfy the definition of a group. 

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