1. 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 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. 2 Group theory Group 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. 3 Group theory Powers: A 1 A 2 A 3 A 4 ……. I (= A 0 ) Suppose A = A π/2 Powers: A 1 A 2 A 3 A 4 ……. I (= A 0 ) Suppose A = A π/2 A 0 A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A 10 A 11 A 12 A 13 A 14 A 15 A 0 A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A 10 A 11 A 12 A 13 A 14 A 15 Cyclical group Infinite? Cyclical group Infinite?

4. 4 Group theory Conjugate products: In general, conjugate products are not = AB ≠ BA BA = A -1 A(BA)AB = B -1 B(AB) = A -1 (AB)A = B -1 (BA)B Conjugate products: In general, conjugate products are not = AB ≠ BA BA = A -1 A(BA)AB = B -1 B(AB) = A -1 (AB)A = B -1 (BA)B

5. 5 Group theory Conjugate products: In general, conjugate products are not = AB ≠ BA BA = A -1 A(BA)AB = B -1 B(AB) = A -1 (AB)A = B -1 (AB)B Thm: Transform of a product by its 1st element is the conjugate product Conjugate products: In general, conjugate products are not = AB ≠ BA BA = A -1 A(BA)AB = B -1 B(AB) = A -1 (AB)A = B -1 (AB)B Thm: Transform of a product by its 1st element is the conjugate product

6. 6 Group theory Conjugate elements: If Y = A -1 XA then X & Y are conjugate elements Conjugate elements: If Y = A -1 XA then X & Y are conjugate elements

7. 7 Group theory Conjugate elements: If Y = A -1 XA 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 Conjugate elements: If Y = A -1 XA 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

8. 8 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 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. 9 Group theory Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Ex: 2/m1, Aπ, m, i What are the subgroups? Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Ex: 2/m1, Aπ, m, i What are the subgroups?

10. 10 Group theory Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Ex: 2/m1, Aπ, m, i What are the subgroups? [1] [1,Aπ] [1,m] [1,i] [1,Aπ,m,i] I and the group itself are, formally, also subgroups Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Ex: 2/m1, Aπ, m, i What are the subgroups? [1] [1,Aπ] [1,m] [1,i] [1,Aπ,m,i] I and the group itself are, formally, also subgroups

11. 11 Group theory Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Notation: Group - Gsubgroup - g B is an "outside" element - in G, but not in g Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Notation: Group - Gsubgroup - g B is an "outside" element - in G, but not in g

12. 12 Group theory Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Notation: Group - Gsubgroup - g B is an "outside" element - in G, but not in g Cosets: g = a 1 a 2 …. A n gB = a 1 B a 2 B …. a n B Bg = Ba 1 Ba 2 …. BA n Elements of cosets must be in G Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Notation: Group - Gsubgroup - g B is an "outside" element - in G, but not in g Cosets: g = a 1 a 2 …. A n gB = a 1 B a 2 B …. a n B Bg = Ba 1 Ba 2 …. BA n Elements of cosets must be in G cosets

13. 13 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: a 1 a 2 ….. a r B 2 ….. B q are all outside elements

14. 14 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: a 1 a 2 ….. a r B 2 ….. B q are all outside elements Then all elements of G are: g = a 1 a 2 ….. a r B 2 g = B 2 a 1 B 2 a 2 ….. B 2 a r B 3 g = B 3 a 1 B 3 a 2 ….. B 3 a r B q g = B q a 1 B q a 2 ….. B q a r 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: a 1 a 2 ….. a r B 2 ….. B q are all outside elements Then all elements of G are: g = a 1 a 2 ….. a r B 2 g = B 2 a 1 B 2 a 2 ….. B 2 a r B 3 g = B 3 a 1 B 3 a 2 ….. B 3 a r B q g = B q a 1 B q a 2 ….. B q a r

15. 15 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 = a 1 a 2 ….. a r B 2 g = B 2 a 1 B 2 a 2 ….. B 2 a r B 3 g = B 3 a 1 B 3 a 2 ….. B 3 a r B q g = B q a 1 B q a 2 ….. B q a r 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 = a 1 a 2 ….. a r B 2 g = B 2 a 1 B 2 a 2 ….. B 2 a r B 3 g = B 3 a 1 B 3 a 2 ….. B 3 a r B q g = B q a 1 B q a 2 ….. B q a r qr elements in G q = index of subgroup g

16. 16 Group theory Subgroups: Ex: g = 1, i (order 2) G = 1, A π, m, i (order 4) B 2 = A π B 3 = m Subgroups: Ex: g = 1, i (order 2) G = 1, A π, m, i (order 4) B 2 = A π B 3 = m

17. 17 Group theory Subgroups: Ex: g = 1, i (order 2) G = 1, A π, m, i (order 4) B 2 = A π B 3 = m g = 1 i B 2 g = A π A π i = A π m B 3 g = m m i = m A π Since B 2 g = B 3 g, g is of index 2 only Subgroups: Ex: g = 1, i (order 2) G = 1, A π, m, i (order 4) B 2 = A π B 3 = m g = 1 i B 2 g = A π A π i = A π m B 3 g = m m i = m A π Since B 2 g = B 3 g, g is of index 2 only

18. 18 Group theory Conjugate subgroups: A in G A -1 g A = hh is also a subgroup Ex: 622 C 2 C 2 C 2 C 2 C 2 C 2 {C 6 } = G 1, C 2 = g A = C 2 Conjugate subgroups: A in G A -1 g A = hh is also a subgroup Ex: 622 C 2 C 2 C 2 C 2 C 2 C 2 {C 6 } = G 1, C 2 = g A = C 2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2

19. 19 Group theory Conjugate subgroups: The set of all conjugate subgroups is called the complete set of conjugates of g Ex: 622 C 2 C 2 C 2 C 2 C 2 C 2 {C 6 } = G 1, C 2 = g A = C 2 1, C 2 = g 1, C 2 = h 1 1, C 2 = h 2 1, C 2 = h 3 1, C 2 = h 4 1, C 2 = h 5 Conjugate subgroups: The set of all conjugate subgroups is called the complete set of conjugates of g Ex: 622 C 2 C 2 C 2 C 2 C 2 C 2 {C 6 } = G 1, C 2 = g A = C 2 1, C 2 = g 1, C 2 = h 1 1, C 2 = h 2 1, C 2 = h 3 1, C 2 = h 4 1, C 2 = h 5 complete set of conjugate subgroups complete set of conjugate subgroups C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2

20. 20 Group theory Invariant subgroups: An invariant subgroup is self conjugate For every B in G B -1 gB = g gB = Bg (right & left cosets =) gB = a 1 B …….. a n B Bg = Ba 1 …….. Ba n Invariant subgroups: An invariant subgroup is self conjugate For every B in G B -1 gB = g gB = Bg (right & left cosets =) gB = a 1 B …….. a n B Bg = Ba 1 …….. Ba n 2 collections of same set of elements 2 collections of same set of elements

21. 21 Group theory Invariant subgroups: Ex: 2/m G = 1, C 2, m, i g = 1, C 2 Invariant subgroups: Ex: 2/m G = 1, C 2, m, i g = 1, C 2

22. 22 Group theory Invariant subgroups: Ex: 2/m G = 1, C 2, m, i g = 1, C 2 1 1 1 = 1 1 C 2 1 = C 2 m -1 C 2 m = C 2 i -1 C 2 i = C 2 Invariant subgroups: Ex: 2/m G = 1, C 2, m, i g = 1, C 2 1 1 1 = 1 1 C 2 1 = C 2 m -1 C 2 m = C 2 i -1 C 2 i = C 2

23. 23 Group theory Invariant subgroups: Every subgroup of index two is invariant G = g, gB G = g, Bg Invariant subgroups: Every subgroup of index two is invariant G = g, gB G = g, Bg

24. 24 Group theory Invariant subgroups: Every subgroup of index two is invariant G = g, gB G = g, Bg Ex: 2/m G = 1, C 2, m, i g = 1, C 2 B = m G = 1, C 2, 1 m, C 2 m = 1, C 2, m, i G = 1, C 2, m 1, m C 2 = 1, C 2, m, i 1 m = m 1 m C 2 = C 2 m Invariant subgroups: Every subgroup of index two is invariant G = g, gB G = g, Bg Ex: 2/m G = 1, C 2, m, i g = 1, C 2 B = m G = 1, C 2, 1 m, C 2 m = 1, C 2, m, i G = 1, C 2, m 1, m C 2 = 1, C 2, m, i 1 m = m 1 m C 2 = C 2 m

25. 25 Group theory Group products: Suppose group g (= a 1 …. a r ) 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 = a 1 …. a r Bg = Ba 1 …. Ba r and Bg = gB (g is of order 2) Group products: Suppose group g (= a 1 …. a r ) 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 = a 1 …. a r Bg = Ba 1 …. Ba r and Bg = gB (g is of order 2)

26. 26 Group theory Group products: Suppose group g (= a 1 …. a r ) 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 = a 1 …. a r Bg = Ba 1 …. Ba r and Bg = gB (g is of order 2) Since g is a group, a i a j = a k ; a k in g Then Ba i a j = Ba k ; Ba k in Bg Products for are Ba i Ba j a i = Ba i B -1 a i B = B a i Group products: Suppose group g (= a 1 …. a r ) 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 = a 1 …. a r Bg = Ba 1 …. Ba r and Bg = gB (g is of order 2) Since g is a group, a i a j = a k ; a k in g Then Ba i a j = Ba k ; Ba k in Bg Products for are Ba i Ba j a i = Ba i B -1 a i B = B a i

27. 27 Group theory Group products: Since g is a group, a i a j = a k ; a k in g Then Ba i a j = Ba k ; Ba k in Bg Products for Bg are Ba i Ba j a i = Ba i B -1 a i B = B a i Ba i Ba j = a i B Ba j B B = I since B is of order 2 Ba i Ba j = a i a j Since B transforms g into itself, a i is an element in g Thus Ba i Ba j with a i a j form a closed set Group products: Since g is a group, a i a j = a k ; a k in g Then Ba i a j = Ba k ; Ba k in Bg Products for Bg are Ba i Ba j a i = Ba i B -1 a i B = B a i Ba i Ba j = a i B Ba j B B = I since B is of order 2 Ba i Ba j = a i a j Since B transforms g into itself, a i is an element in g Thus Ba i Ba j with a i a j form a closed set

28. 28 Group theory Group products: Identity is in g Inverses - a n in g (Ba n ) -1 = a n B -1 = B -1 (a n ) = B (a n ) in Bg Therefore g, Bg is a group Group products: Identity is in g Inverses - a n in g (Ba n ) -1 = a n B -1 = B -1 (a n ) = B (a n ) in Bg Therefore g, Bg is a group

29. 29 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 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