Understanding Space Groups: Quotients, Homomorphisms & Translations

1. Space groups Translations T = ut1(1-D) u is an integer The set of all lattice vectors is a group (the set of all integers (±) is a group)

2. Space groups Quotient groups Thm: Cosets of an invariant subgroup form a group in which the subgroup is the identity element G = group, g = invariant subgroup, Bi = outside elements Then G, expanded into its cosets, is g, B2g, B3g ……

3. Space groups Quotient groups Thm: Cosets of an invariant subgroup form a group in which the subgroup is the identity element G = group, g = invariant subgroup, Bi = outside elements Then G, expanded into its cosets, is g, B2g, B3g …… g = a1 a2 …. (B1 = a1 = 1) B2g = B2 B2a2 …. B3g = B3 B3a2 ….

4. Space groups Quotient groups Products: (Bi g)(Bj g) = Bi Bj g g = Bi Bj g (Bi g = g Bi) Bi, Bj in G & thus Bi Bj = Bk ap (Bi g)(Bj g) = Bi Bj g = Bk ap g ap g = g --> Bi Bj g = Bk g (closed set)

5. Space groups Quotient groups Identity: g g = g (Bi g) g = Bi g (g is identity element) Inverses: (Bi g) (Bi g) = Bi g Bi g = Bi Bi g g = g inverse of Bi g is Bi g Thus, cosets form a group, called G/g, the quotient group -1 -1 -1 -1

6. Space groups Homomorphism Suppose in the quotient group have product (Bi ar) (Bj as) g invariant subgroup any B transforms g into itself any B-1 a B is in g

7. Space groups Homomorphism Suppose in the quotient group have product (Bi ar) (Bj as) g invariant subgroup any B transforms g into itself any B-1 a B is in g Bj ar Bj = at ar Bj =Bj at Then (Bi ar) (Bj as) = Bi ar Bj as = Bi Bj at as = Bk ap at as = Bk aq Product is in same coset Bk as product of its Bs -1

8. Space groups Homomorphism Now: G G/g a1 a2 ……….. ar g B2a1 B2a2 ……….. B2ar B2g B3a1 B3a2 ……….. B3ar B3g Bsa1 Bsa2 ……….. Bsar Bsg

9. Space groups Homomorphism Now: G G/g a1 a2 ……….. ar g B2a1 B2a2 ……….. B2ar B2g B3a1 B3a2 ……….. B3ar B3g Bsa1 Bsa2 ……….. Bsar Bsg Any of the r elements in a row of G corresponds to one element in G/g

10. Space groups Homomorphism Now: G G/g a1 a2 ……….. ar g B2a1 B2a2 ……….. B2ar B2g B3a1 B3a2 ……….. B3ar B3g Bsa1 Bsa2 ……….. Bsar Bsg Any of the r elements in a row of G corresponds to one element in G/g Any of the products in G of elements in Bith w/ elements in Bjth row gives a product in the Bkth row

11. Space groups Homomorphism Now: G G/g a1 a2 ……….. ar g B2a1 B2a2 ……….. B2ar B2g B3a1 B3a2 ……….. B3ar B3g Bsa1 Bsa2 ……….. Bsar Bsg Any of the r elements in a row of G corresponds to one element in G/g Any of the products in G of elements in Bith w/ elements in Bjth row gives a product in the Bkth row Only r products in kth row, so any of these products corresponds to one product in G/g

12. Space groups Homomorphism In summary: G G/g r elements Bi correspond to one element Big r elements Bj correspond to one element Bjg r products Bi Bj correspond to one product Bi Bj g The two groups are HOMOMORPHIC

13. Space groups Infinite groups Translation groups have n = ∞ Any group having a translation component has n = ∞ Ex: mt (glide) 1 mt mt mt ……… 1 mt m2t2 m3t3……… 2 3

14. Space groups Infinite groups Translation groups have n = ∞ Any group having a translation component has n = ∞ Ex: mt (glide) 1 mt mt mt ……… 1 mt m2t2 m3t3……… m2 =1 1 mt t2 mt3……… 2 3

15. Space groups Infinite groups Translation groups have n = ∞ Any group having a translation component has n = ∞ Ex: mt (glide) 1 mt mt mt ……… 1 mt m2t2 m3t3……… m2 =1 1 mt t2 mt3……… t2 = T 1 mt T mtT……… 2 3

16. Space groups Infinite groups t2 = T 1 mt T mtT……… or: 1 T T2 T3……… mt mtT mtT2 mtT3……… Group mt has the subgroup consisting of the powers of T (invariant, ∞)

17. Space groups Infinite groups The group mt is homorphic w/ the group m mt m 1 T T2 T3………1 mt mtT mtT2 mtT3………m

18. Space groups Ex: 41 41 4 1 T T2 T3………1 C4t C4tT C4tT2 C4tT3………C4 C4t C4tT C4tT2 C4tT3………C4 C4t C4tT C4tT2 C4tT3………C4 A group for glide planes or screw axes is homomorphic w/ the corresponding isogonal pt. grp. 2 2 2 2 2 3 3 3 3 3

19. Space groups Ex: 41 Also: GC4t C4t C4t form a group (G= any operation in 1, T, T2, T3………) 2 2 3 3

20. Space groups If the mult. table of a pt. grp. is known, mult. table of corresponding quotient grp. is known and the same

21. Space groups If the mult. table of a pt. grp. is known, mult. table of corresponding quotient grp. is known and the same The rotational component in a coset in the space grp. & the pt. grp are the same