1 Group theory 1st postulate - combination of any 2 elements, including an element w/ itself, is a member of the group. 2nd postulate - the set of elements

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Group theoryGroup theory

1st postulate - combination of any 2 elements, including an element w/ itself, is a member of the group.

2nd postulate - the set of elements of the group contains the identity element (IA = A)

3rd postulate - for each element A, there is a unique element A' which is the inverse of A (AA-1 = I)

1st postulate - combination of any 2 elements, including an element w/ itself, is a member of the group.

2nd postulate - the set of elements of the group contains the identity element (IA = A)

3rd postulate - for each element A, there is a unique element A' which is the inverse of A (AA-1 = I)

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Group multiplication tables - example: 2/mGroup multiplication tables - example: 2/m

1 Aπ m i

1 1 Aπ m i

Aπ Aπ 1 i m

m m i 1 Aπ

i i m Aπ 1

1 Aπ m i

1 1 Aπ m i

Aπ Aπ 1 i m

m m i 1 Aπ

i i m Aπ 1

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Powers:

A1 A2 A3 A4 ……. I (= A0)

Suppose A = Aπ/2

Powers:

A1 A2 A3 A4 ……. I (= A0)

Suppose A = Aπ/2

A0 A1 A2 A3

A4 A5 A6 A7

A8 A9 A10 A11

A12 A13 A14 A15

A0 A1 A2 A3

A4 A5 A6 A7

A8 A9 A10 A11

A12 A13 A14 A15

Cyclical group

Infinite?

Cyclical group

Infinite?

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Conjugate products:

In general, conjugate products are not =

AB ≠ BA

BA = A-1A(BA) AB = B-1B(AB) = A-1(AB)A = B-1(BA)B

Conjugate products:


AB ≠ BA

BA = A-1A(BA) AB = B-1B(AB) = A-1(AB)A = B-1(BA)B

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Group theory

Conjugate products:


AB ≠ BA

BA = A-1A(BA) AB = B -1B(AB) = A-1(AB)A = B -1(AB)B

Thm: Transform of a product by its1st element is the conjugate product

Conjugate products:


AB ≠ BA

BA = A-1A(BA) AB = B -1B(AB) = A-1(AB)A = B -1(AB)B

Thm: Transform of a product by its1st element is the conjugate product

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Conjugate elements:

If Y = A-1XA then X & Y are conjugate elements

Conjugate elements:


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Group theory

Conjugate elements:


Sets of conjugate elements:

Ex - in point group 322, 2-fold axes 120° apart & 3-fold axis

these three 2-fold axes form a set of conjugate elementswrt the 3-fold axis

Conjugate elements:


Sets of conjugate elements:

Ex - in point group 322, 2-fold axes 120° apart & 3-fold axis

these three 2-fold axes form a set of conjugate elementswrt the 3-fold axis

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Invariant elements:

If every element of a group transforms a particular element of that group into itself, then that element is invariant

Ex: 6-fold axis in 6/m

m takes 6 into itself

Invariant elements:

If every element of a group transforms a particular element of that group into itself, then that element is invariant

Ex: 6-fold axis in 6/m

m takes 6 into itself

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Subgroups:

A smaller collection of elements from a group that isitself a group is a subgroup

Ex: 2/m 1, Aπ, m, i

What are the subgroups?

Subgroups:


Ex: 2/m 1, Aπ, m, i

What are the subgroups?

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Group theory

Subgroups:


Notation:

Group - G subgroup - g

B is an "outside" element - in G, but not in g

Subgroups:


Notation:



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Subgroups:


Notation:



Cosets: g = a1 a2 …. An

gB = a1B a2B …. anB

Bg = Ba1 Ba2 …. BAn

Elements of cosets must be in G

Subgroups:


Notation:



Cosets: g = a1 a2 …. An

gB = a1B a2B …. anB

Bg = Ba1 Ba2 …. BAn

Elements of cosets must be in G

cosetscosets

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Subgroups:

Thm: The order of a subgroup is a factor of the order of the group. (order = # elements in g, or G)

r elements of g: a1 a2 ….. ar

B2 ….. Bq are all outside elements

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Subgroups:




Then all elements of G are:

g = a1 a2 ….. ar

B2g = B2a1 B2a2 ….. B2ar

B3g = B3a1 B3a2 ….. B3ar

Bqg = Bqa1 Bqa2 ….. Bqar

Subgroups:





g = a1 a2 ….. ar

B2g = B2a1 B2a2 ….. B2ar

B3g = B3a1 B3a2 ….. B3ar


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Subgroups:



g = a1 a2 ….. ar

B2g = B2a1 B2a2 ….. B2ar

B3g = B3a1 B3a2 ….. B3ar


qr elements in G q = index of subgroup g

Subgroups:



g = a1 a2 ….. ar

B2g = B2a1 B2a2 ….. B2ar

B3g = B3a1 B3a2 ….. B3ar


qr elements in G q = index of subgroup g

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Subgroups:

Ex: g = 1, i (order 2) G = 1, Aπ, m, i (order 4) B2 = Aπ B3 = m

Subgroups:


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Subgroups:


g = 1 i

B2g = Aπ Aπ i = Aπ m

B3g = m m i = m Aπ

Since B2g = B3g, g is of index 2 only

Subgroups:


g = 1 i

B2g = Aπ Aπ i = Aπ m

B3g = m m i = m Aπ

Since B2g = B3g, g is of index 2 only

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Conjugate subgroups:

A in G A-1 g A = h h is also a subgroup

Ex: 622 C2 C2 C2 C2 C2 C2 {C6} = G

1, C2 = g A = C2


A in G A-1 g A = h h is also a subgroup

Ex: 622 C2 C2 C2 C2 C2 C2 {C6} = G

1, C2 = g A = C2

C2C2

C2C2

C2C2

C2C2

C2C2

C2C2

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The set of all conjugate subgroups is called the complete set of conjugates of g

Ex: 622 C2 C2 C2 C2 C2 C2 {C6} = G

1, C2 = g A = C2

1, C2 = g1, C2 = h1

1, C2 = h2

1, C2 = h3

1, C2 = h4

1, C2 = h5


The set of all conjugate subgroups is called the complete set of conjugates of g

Ex: 622 C2 C2 C2 C2 C2 C2 {C6} = G

1, C2 = g A = C2

1, C2 = g1, C2 = h1

1, C2 = h2

1, C2 = h3

1, C2 = h4

1, C2 = h5

complete setof conjugatesubgroups

complete setof conjugatesubgroups C2C2

C2C2

C2C2

C2C2

C2C2

C2C2

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Invariant subgroups:

An invariant subgroup is self conjugate

For every B in G

B-1gB = g

gB = Bg

(right & left cosets =)

gB = a1B …….. anBBg = Ba1 …….. Ban


An invariant subgroup is self conjugate

For every B in G

B-1gB = g

gB = Bg

(right & left cosets =)

gB = a1B …….. anBBg = Ba1 …….. Ban

2 collections of sameset of elements2 collections of sameset of elements

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Ex: 2/m G = 1, C2, m, i

g = 1, C2


Ex: 2/m G = 1, C2, m, i

g = 1, C2

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Ex: 2/m G = 1, C2, m, i

g = 1, C2

1 1 1 = 1

1 C2 1 = C2

m-1 C2 m = C2

i-1 C2 i = C2


Ex: 2/m G = 1, C2, m, i

g = 1, C2

1 1 1 = 1

1 C2 1 = C2

m-1 C2 m = C2

i-1 C2 i = C2

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Every subgroup of index two is invariant

G = g, gBG = g, Bg



G = g, gBG = g, Bg

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G = g, gBG = g, Bg

Ex: 2/m G = 1, C2, m, i g = 1, C2 B = m

G = 1, C2, 1 m, C2 m = 1, C2, m, i

G = 1, C2, m 1, m C2 = 1, C2, m, i

1 m = m 1m C2 = C2 m



G = g, gBG = g, Bg

Ex: 2/m G = 1, C2, m, i g = 1, C2 B = m

G = 1, C2, 1 m, C2 m = 1, C2, m, i

G = 1, C2, m 1, m C2 = 1, C2, m, i

1 m = m 1m C2 = C2 m

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Group products:

Suppose group g (= a1 …. ar)B not in g

Thm: if element B of order 2 transforms group g into itself then elements in g and Bg form a group

g = a1 …. ar Bg = Ba1 …. Bar

and Bg = gB (g is of order 2)

Group products:



g = a1 …. ar Bg = Ba1 …. Bar


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Group products:



g = a1 …. ar Bg = Ba1 …. Bar


Since g is a group, ai aj = ak; ak in g

Then Bai aj = Bak; Bak in Bg

Products for are Bai Baj

ai = Bai B-1 ai B = B ai

Group products:



g = a1 …. ar Bg = Ba1 …. Bar




Products for are Bai Baj

ai = Bai B-1 ai B = B ai

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Group products:



Products for Bg are Bai Baj

ai = Bai B-1ai B = B ai

Bai Baj = ai B Baj

B B = I since B is of order 2

Bai Baj = ai aj

Since B transforms g into itself, ai is an element in g

Thus Bai Baj with ai aj form a closed set

Group products:



Products for Bg are Bai Baj

ai = Bai B-1ai B = B ai

Bai Baj = ai B Baj

B B = I since B is of order 2

Bai Baj = ai aj

Since B transforms g into itself, ai is an element in g

Thus Bai Baj with ai aj form a closed set

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Group products:

Identity is in g

Inverses -an in g

(Ban)-1 = an B-1 = B-1 (an ) = B (an ) in Bg

Therefore g, Bg is a group

Group products:

Identity is in g

Inverses -an in g

(Ban)-1 = an B-1 = B-1 (an ) = B (an ) in Bg

Therefore g, Bg is a group

-1

-1 -1 -1

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Group products:

Extended arguments give

Thm: If g & h two groups w/ no common element except I

If each element of h transforms g into itself

Then the set of products of g & h form a group

Group products:

Extended arguments give

Thm: If g & h two groups w/ no common element except I

If each element of h transforms g into itself

Then the set of products of g & h form a group

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1 Group theory 1st postulate - combination of any 2 elements, including an element w/ itself, is a member of the group. 2nd postulate - the set of elements