08 group theory

8/3/2019 08 Group Theory

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Group Theory

Point Groups


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Point Group

A collection of symmetry elements obeying the properties of

a mathematical group and having one point in common,

which remains fixed through all symmetry operations.

Point groups describe the allowable collection of symmetries

that 3-dimensional objects may posses.

The collection of symmetry operators comprising a point

group follow the rules of group theory.


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Group

A collection of elements which is closed under single

valued associative binary operations, which contain a

single element satisfying the identity law, and whichpossesses a reciprocal element for each element in the

collection.


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CollectionA specified number of objects.

ElementsThe constituents of the group. These objects may be

abstract and only given symbolic names.

Binary OperationBasic law which describes the combination of elements.

The combination is defined for only two elements at a time.

Addition and multiplication in algebra Successive application of symmetry operations to an

object


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Single-ValuedThe combination of two elements yields a unique result.

ClosedThe combination of any two group elements must yield

another element in the group.

Identity LawThere must be an element in the group that when combined

with all other elements in the group leaves each unchanged.

This element is called the identityorunit element. It

commutes with all elements in the group.

EA = AE = A


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Commutative PropertyThough an element which commutes is required as part of

the group in generalAB BA

Commuting groups are called abelian.

ReciprocalFor each element in the group there must be another element

in the group, A-1

, such thatAA-1 = A-1A = E

AssociativeThe associative law of combination must hold.

(AB)C = A(BC)


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Building a Group

Start with three symmetry elements:

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100

010

001

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010

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Identity 4-fold (41) Inversion


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Generate a Member

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Generate Another

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1

4C

1

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1

2C


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Generate Another

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Generate Two More

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4

C1

2

C3

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Summary

The following are then the elements of a closed group:

How can we be sure? Must combine all the operators

to produce a multiplication table and show that no new

operators appear.


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hCm 44

08 group theory

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