1 group theory 1st postulate - combination of any 2 elements, including an element w/ itself, is a...
Post on 18-Dec-2015
216 Views
Preview:
TRANSCRIPT
1
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)
2
Group theoryGroup theory
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
3
Group theoryGroup theory
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?
4
Group theoryGroup theory
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:
In general, conjugate products are not =
AB ≠ BA
BA = A-1A(BA) AB = B-1B(AB) = A-1(AB)A = B-1(BA)B
5
Group theory
Conjugate products:
In general, conjugate products are not =
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:
In general, conjugate products are not =
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
6
Group theoryGroup theory
Conjugate elements:
If Y = A-1XA then X & Y are conjugate elements
Conjugate elements:
If Y = A-1XA then X & Y are conjugate elements
7
Group theory
Conjugate elements:
If Y = A-1XA then X & Y are 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:
If Y = A-1XA then X & Y are 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
8
Group theoryGroup theory
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
9
Group theoryGroup theory
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:
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?
11
Group theory
Subgroups:
A smaller collection of elements from a group that isitself a group is a subgroup
Notation:
Group - G subgroup - g
B is an "outside" element - in G, but not in g
Subgroups:
A smaller collection of elements from a group that isitself a group is a subgroup
Notation:
Group - G subgroup - g
B is an "outside" element - in G, but not in g
12
Group theoryGroup theory
Subgroups:
A smaller collection of elements from a group that isitself a group is a subgroup
Notation:
Group - G subgroup - g
B is an "outside" element - in G, but not in g
Cosets: g = a1 a2 …. An
gB = a1B a2B …. anB
Bg = Ba1 Ba2 …. BAn
Elements of cosets must be in G
Subgroups:
A smaller collection of elements from a group that isitself a group is a subgroup
Notation:
Group - G subgroup - g
B is an "outside" element - in G, but not in g
Cosets: g = a1 a2 …. An
gB = a1B a2B …. anB
Bg = Ba1 Ba2 …. BAn
Elements of cosets must be in G
cosetscosets
13
Group theoryGroup theory
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
14
Group theoryGroup theory
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
Then all elements of G are:
g = a1 a2 ….. ar
B2g = B2a1 B2a2 ….. B2ar
B3g = B3a1 B3a2 ….. B3ar
Bqg = Bqa1 Bqa2 ….. Bqar
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
Then all elements of G are:
g = a1 a2 ….. ar
B2g = B2a1 B2a2 ….. B2ar
B3g = B3a1 B3a2 ….. B3ar
Bqg = Bqa1 Bqa2 ….. Bqar
15
Group theoryGroup theory
Subgroups:
Thm: The order of a subgroup is a factor of the order of the group. (order = # elements in g, or G)
Then all elements of G are:
g = a1 a2 ….. ar
B2g = B2a1 B2a2 ….. B2ar
B3g = B3a1 B3a2 ….. B3ar
Bqg = Bqa1 Bqa2 ….. Bqar
qr elements in G q = index of subgroup g
Subgroups:
Thm: The order of a subgroup is a factor of the order of the group. (order = # elements in g, or G)
Then all elements of G are:
g = a1 a2 ….. ar
B2g = B2a1 B2a2 ….. B2ar
B3g = B3a1 B3a2 ….. B3ar
Bqg = Bqa1 Bqa2 ….. Bqar
qr elements in G q = index of subgroup g
16
Group theoryGroup theory
Subgroups:
Ex: g = 1, i (order 2) G = 1, Aπ, m, i (order 4) B2 = Aπ B3 = m
Subgroups:
Ex: g = 1, i (order 2) G = 1, Aπ, m, i (order 4) B2 = Aπ B3 = m
17
Group theoryGroup theory
Subgroups:
Ex: g = 1, i (order 2) G = 1, Aπ, m, i (order 4) B2 = Aπ B3 = m
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:
Ex: g = 1, i (order 2) G = 1, Aπ, m, i (order 4) B2 = Aπ B3 = m
g = 1 i
B2g = Aπ Aπ i = Aπ m
B3g = m m i = m Aπ
Since B2g = B3g, g is of index 2 only
18
Group theoryGroup theory
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
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
C2C2
C2C2
C2C2
C2C2
C2C2
C2C2
19
Group theoryGroup theory
Conjugate subgroups:
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
Conjugate subgroups:
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
20
Group theoryGroup theory
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
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
2 collections of sameset of elements2 collections of sameset of elements
21
Group theoryGroup theory
Invariant subgroups:
Ex: 2/m G = 1, C2, m, i
g = 1, C2
Invariant subgroups:
Ex: 2/m G = 1, C2, m, i
g = 1, C2
22
Group theoryGroup theory
Invariant subgroups:
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
Invariant subgroups:
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
23
Group theoryGroup theory
Invariant subgroups:
Every subgroup of index two is invariant
G = g, gBG = g, Bg
Invariant subgroups:
Every subgroup of index two is invariant
G = g, gBG = g, Bg
24
Group theoryGroup theory
Invariant subgroups:
Every subgroup of index two is invariant
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
Invariant subgroups:
Every subgroup of index two is invariant
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
25
Group theoryGroup theory
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:
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)
26
Group theoryGroup theory
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)
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:
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)
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
27
Group theoryGroup theory
Group products:
Since g is a group, ai aj = ak; ak in g
Then Bai aj = Bak; Bak in Bg
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:
Since g is a group, ai aj = ak; ak in g
Then Bai aj = Bak; Bak in Bg
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
28
Group theoryGroup theory
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
29
Group theoryGroup theory
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
top related