Group theory

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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 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 theory
Group multiplication tables - example 2/m
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Group theory
Powers A1 A2 A3 A4 . I (
A0) Suppose A Ap/2

• A0 A1 A2 A3
• A4 A5 A6 A7
• A8 A9 A10 A11
• A12 A13 A14 A15

Cyclical group Infinite?
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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(BA)B
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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
its 1st element is the conjugate product
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Group theory
Conjugate elements If Y A-1XA then X
Y are conjugate elements
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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
elements wrt the 3-fold axis
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Group 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
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Group theory
Subgroups A smaller collection of elements
from a group that is itself a group is a
subgroup Ex 2/m 1, Ap, m, i What are
the subgroups?
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Group theory
Subgroups A smaller collection of elements
from a group that is itself a group is a
subgroup Ex 2/m 1, Ap, m, i What are
the subgroups? 1 1,Ap 1,m 1,i
1,Ap,m,i I and the group itself are,
formally, also subgroups
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Group theory
Subgroups A smaller collection of elements
from a group that is itself a group is a
subgroup Notation Group - G subgroup -
g B is an "outside" element - in G, but not
in g
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Group theory
Subgroups A smaller collection of elements
from a group that is itself 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
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Group 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
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Group 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
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Group 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
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Group theory
Subgroups Ex g 1, i (order 2) G
1, Ap, m, i (order 4) B2 Ap B3 m

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Group theory
Subgroups Ex g 1, i (order 2) G
1, Ap, m, i (order 4) B2 Ap B3 m
g 1 i B2g Ap Ap
i Ap m B3g m m i
m Ap Since B2g B3g, g is of index
2 only
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Group theory
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Group theory
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Group theory
Invariant subgroups Ex 2/m G 1, C2,
m, i g 1, C2
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Group theory
Invariant subgroups Every subgroup of index
two is invariant G g, gB G g, Bg
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Group 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)
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Group 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
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Group 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-1 ai 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 theory
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Group 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