Symmetric Gro up pe rmutatio ns o f 3 o bje c ts Gro up elements c - - PowerPoint PPT Presentation

Symmetric Gro up

 pe rmutatio ns o f 3 o bje c ts  Gro up elements c an be written in this fo rmat:

Symmetric Gro up

No te

Symmetric group

 Product operation  Is the group elements {e,a,b,ab,b,b2} isomorphic

to the above permutation elements?

Symmetric group

 Order of , which is a symmetric group involving

permutation of n objects, is n!



is called symmetric group of degree n

 Subgroups of are called permutation groups  Cayley’s theorem states that every finite group is

isomorphic to a permutation group embedded inside

 Any permutation element can be equivalently

represented as a product of disjoint permutation cycles

Symmetric group

 Consider the following permutation element  This can be written in the following disjoint cycle

structure

 Cycle decomposition is useful for multiplication of two

permutation elements

Symmetric Gro up

Two-cycle is called transposition. Inverse of the transposition is the same element. Inverse of 3-cycle (123) is (132). Why? Every n-cycle can be written as product of transpositions

Symmetric Gro up

Note that the product of the two permutation elements have six-cycle structure. Of course the elements are different.

Symmetric Gro up

Depending on the odd or even number of transpositions, permutation element is called odd or even permutation Any k-cycle can be broken into products of transpositions (2-cycle)

Symmetric group

 Any permutation element will have where

where k runs from 1 to n such that

 All permutation elements with the above cycle structure

can be shown to be conjugate elements ( prove)

 Total number of permutation elements( within the

conjugacy class given by the cycle structure) is

Symmetric group

 The number of conjugacy classes in the symmetric group is

equal to the number of ways of partitioning integer n

 For example, n=5 can be broken into 7 distinct conjugacy

classes

 Convenient way of diagrammatically representing the

conjugacy classes using Young diagrams

 1-cycles by single box, 2-cycle by double vertical box and so

• n

 Identity element for n=5 is five 1-cycles denoted by

Symmetric group

 Product of two 2-cycles and one 1-cycle will be represented

by

 One 5-cycle will be

Symmetric group

 Set of even permutation elements form a group known as alternating group  Conjugate elements of even permutation elements will always

be even which implies



is an invariant or normal subgroup

 Factor group  Show that there are only two cosets possible or the factor

group has only two elements [e, (1,2)]

Direct Product groups

 For two groups, direct product group is  Example  Note that the elements of both the groups commute

and order of G is product of order of the two groups

Semi-Direct product groups

 Let K be invariant subgroup of G and T be another

subgroup of G such that identity element is the only common element between K and T

 Then, G is the semi-direct product group denoted by  Show that T are coset elements  Example

Symmetry of a molecule

 Rotations and reflections which leaves the molecule

invariant

 Axis of rotational symmetry  Plane of symmetry- two types  Plane perpendicular to axis (horizontal mirror plane)-  Plane containing the axis (vertical mirror plane)-  Roto-reflection symmetry-  There could be diagonal plane of symmetry (cube)-

Improper symmetry

• perations

Mirror planes =>

h => mirror plane perpendicular to a principal axis of rotation v => mirror plane containing principal axis of rotation d => mirror plane bisects dihedral angle made by the principal axis of rotation and two adjacent C2 axes perpendicular to principal rotation axis

Rotoreflection

Improper axis of rotation => Sn

 rotation about n axis followed by reflection about the

plane of symmetry (check it generates abelian group)

Point Groups

 The set containing elementary operations plus various

symmetry operations as a result of composing elementary operations forms a group called Point group.

 At least one atom in the molecule is fixed under the

symmetry operations- hence the name point group

 Number of elements in the point group is finite  By Cayley’s theorem, point groups(symmetry of non-

linear molecule) are isomorphic to subgroups of symmetry group

Schoenflies Notation

Water molecule Symmetry

σv(xz) C2

Group symmetry is

Ammonia Molecule

Group symmetry?

Methane

Group symmetry?

Streographic projection

Streographic projection

Streographic projection

Streographic projection

C3 C3v D3 D2h

Symmetries of a cube

Embed tetrahedron in cube

Tetrahedral molecule

C2 Three C2 C3 Four 3-fold axes Pure Rotations give group T

Including reflections

σ is the mirror (reflection) plane six mirror reflection planes (6σ) S4 is a rotation by 90° followed by a mirror reflection a tetrahedral Structure has total 24 symmetry operations

Representation of