1 Space groups Translations T = ut 1 (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, B i = outside elements Then G, expanded into its cosets, is g, B 2 g, B 3 g ……

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, B i = outside elements Then G, expanded into its cosets, is g, B 2 g, B 3 g …… g = a 1 a 2 …. (B 1 = a 1 = 1) B 2 g = B 2 B 2 a 2 …. B 3 g = B 3 B 3 a 2 ….

4 Space groups Quotient groups Products: (B i g)(B j g) = B i B j g g = B i B j g (B i g = g B i ) B i, B j in G & thus B i B j = B k a p (B i g)(B j g) = B i B j g = B k a p g a p g = g --> B i B j g = B k g (closed set)

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

6 Space groups Homomorphism Suppose in the quotient group have product (B i a r ) (B j a s ) 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 (B i a r ) (B j a s ) g invariant subgroup any B transforms g into itself any B -1 a B is in g B j a r B j = a t a r B j = B j a t Then (B i a r ) (B j a s ) = B i a r B j a s = B i B j a t a s = B k a p a t a s = B k a q Product is in same coset B k as product of its Bs

8 Space groups Homomorphism Now: GG/g a 1 a 2 ……….. a r g B 2 a 1 B 2 a 2 ……….. B 2 a r B 2 g B 3 a 1 B 3 a 2 ……….. B 3 a r B 3 g B s a 1 B s a 2 ……….. B s a r B s g

9 Space groups Homomorphism Now: GG/g a 1 a 2 ……….. a r g B 2 a 1 B 2 a 2 ……….. B 2 a r B 2 g B 3 a 1 B 3 a 2 ……….. B 3 a r B 3 g B s a 1 B s a 2 ……….. B s a r B s g Any of the r elements in a row of G corresponds to one element in G/g

10 Space groups Homomorphism Now: GG/g a 1 a 2 ……….. a r g B 2 a 1 B 2 a 2 ……….. B 2 a r B 2 g B 3 a 1 B 3 a 2 ……….. B 3 a r B 3 g B s a 1 B s a 2 ……….. B s a r B s g 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 B i th w/ elements in B j th row gives a product in the B k th row

11 Space groups Homomorphism Now: GG/g a 1 a 2 ……….. a r g B 2 a 1 B 2 a 2 ……….. B 2 a r B 2 g B 3 a 1 B 3 a 2 ……….. B 3 a r B 3 g B s a 1 B s a 2 ……….. B s a r B s g 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 B i th w/ elements in B j th row gives a product in the B k th row Only r products in k th row, so any of these products corresponds to one product in G/g

12 Space groups Homomorphism In summary: GG/g r elements B i correspond toone element B i g r elements B j correspond toone element B j g r products B i B j correspond toone product B i B j g The two groups are HOMOMORPHIC

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

14 Space groups Infinite groups Translation groups have n = ∞ Any group having a translation component has n = ∞ Ex:m t (glide) 1 m t m t m t ……… 1 mt m 2 t 2 m 3 t 3 ……… m 2 = 1 1 mt t 2 mt 3 ……… 2 3

15 Space groups Infinite groups Translation groups have n = ∞ Any group having a translation component has n = ∞ Ex:m t (glide) 1 m t m t m t ……… 1 mt m 2 t 2 m 3 t 3 ……… m 2 = 1 1 mt t 2 mt 3 ……… t 2 = T 1 mt T mtT ……… 2 3

16 Space groups Infinite groups t 2 = T 1 mt T mtT ……… or: 1 T T 2 T 3 ……… mt mtT mtT 2 mtT 3 ……… Group m t has the subgroup consisting of the powers of T (invariant, ∞ )

17 Space groups Infinite groups The group m t is homorphic w/ the group m m t m 1 T T 2 T 3 ………1 mt mtT mtT 2 mtT 3 ………m

18 Space groups Ex: T T 2 T 3 ………1 C 4 t C 4 tT C 4 tT 2 C 4 tT 3 ………C 4 A group for glide planes or screw axes is homomorphic w/ the corresponding isogonal pt. grp

19 Space groups Ex: 4 1 Also:   C 4 t C 4 t C 4 t form a group (   = any operation in 1, T, T 2, T 3 ………) 2332

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