SLIDE 1
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 - - 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,b^2} isomorphic to the above
SLIDE 2
SLIDE 3
Symmetric group
Product operation Is the group elements {e,a,b,ab,b,b^2} isomorphic
to the above permutation elements?
SLIDE 4
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
SLIDE 5
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
SLIDE 6
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
SLIDE 7
Symmetric Gro up
Note that the product of the two permutation elements have six-cycle structure. Of course the elements are different.
SLIDE 8
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)
SLIDE 9
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
SLIDE 10
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
SLIDE 11
Symmetric group
Product of two 2-cycles and one 1-cycle will be represented
by
One 5-cycle will be
SLIDE 12
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)]
SLIDE 13
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
SLIDE 14
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
SLIDE 15
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)-
SLIDE 16
Water molecule Symmetry
σv(xz) C2
SLIDE 17
Ammonia Molecule
Group symmetry?
SLIDE 18
Methane
Group symmetry?
SLIDE 19
Streographic projection
SLIDE 20
Streographic projection
SLIDE 21
Streographic projection
SLIDE 22
SLIDE 23