Group theory

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)

2. Group theory 1 Aπ m i 1 1 Aπ m i Aπ Aπ 1 i m m m i 1 Aπ i i m Aπ 1 Group multiplication tables - example: 2/m

3. Group theory 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 Cyclical group Infinite?

4. 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

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 its 1st element is the conjugate product

6. Group theory 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 elements wrt the 3-fold axis

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

9. Group theory Subgroups: A smaller collection of elements from a group that is itself a group is a subgroup Ex: 2/m 1, Aπ, m, i What are the subgroups?

10. 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

11. Group theory cosets 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

12. 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

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: 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

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) 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

15. Group theory • Subgroups: • Ex: g = 1, i (order 2) G = 1, Aπ, m, i (order 4) • B2 = Aπ B3 = m

16. Group 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

17. Group 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 C2 C2 C2 C2 C2 C2

18. Group 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 = g 1, C2 = h1 1, C2 = h2 1, C2 = h3 1, C2 = h4 1, C2 = h5 C2 C2 C2 complete set of conjugate subgroups C2 C2 C2

19. Group theory Invariant subgroups: An invariant subgroup is self conjugate For every B in G B-1gB = g gB = Bg (right & left cosets =) gB = a1B …….. anB Bg = Ba1 …….. Ban 2 collections of same set of elements

20. Group theory Invariant subgroups: Ex: 2/m G = 1, C2, m, i g = 1, C2

21. Group 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

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

23. Group theory Invariant subgroups: Every subgroup of index two is invariant G = g, gB G = 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 1 m C2 = C2 m

24. 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)

25. 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

26. 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

27. Group 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 -1 -1 -1 -1

28. 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