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Point Symmetry

Point Symmetry. Groups w/ improper operations Any group containing 2nd sort operations also contains 1st sort operations (A a i) (B a i) = A a B a i i = A a B a. Point Symmetry. Groups w/ improper operations Any group containing 2nd sort operations also

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Point Symmetry

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  1. Point Symmetry Groups w/ improper operations Any group containing 2nd sort operations also contains 1st sort operations (Aa i) (Ba i) = Aa Ba i i = Aa Ba

  2. Point Symmetry Groups w/ improper operations Any group containing 2nd sort operations also contains 1st sort operations (Aa i) (Ba i) = Aa Ba i i = Aa Ba 1st sort operations in such a group G form a subgroup of index 2 g = 1 a2 …. an Bg = B Ba2 …. Ban B is any 2nd sort operation in G

  3. Point Symmetry Groups w/ improper operations Any group containing 2nd sort operations also contains 1st sort operations (Aa i) (Ba i) = Aa Ba i i = Aa Ba 1st sort operations in such a group G form a subgroup of index 2 g = 1 a2 …. an Bg = B Ba2 …. Ban B is any 2nd sort operation in G (Ba i) ar = (Ba ar) i all 2nd sort operations in Bg 1st sort

  4. Point Symmetry Groups w/ improper operations 1. Any group of 2nd sort contains equal nos. of operations of 1st & 2nd sort 2. In any group of the 2nd sort, 1st sort operations form a subgroup of index 2

  5. Point Symmetry Groups w/ improper operations 1. Any group of 2nd sort contains equal nos. of operations of 1st & 2nd sort 2. In any group of the 2nd sort, 1st sort operations form a subgroup of index 2 3. A group of the 2nd sort can be formed by adding any second sort operation (and its products) to a group such that it transforms that group into itself

  6. Point Symmetry Groups w/ improper operations 1. Any group of 2nd sort contains equal nos. of operations of 1st & 2nd sort 2. In any group of the 2nd sort, 1st sort operations form a subgroup of index 2 3. A group of the 2nd sort can be formed by adding any second sort operation (and its products) to a group such that it transforms that group into itself 4. All groups of the 2nd sort can be formed by starting with all groups of the 1st sort & adding each possible 2nd sort operation (i or m) & products which transforms the group into itself

  7. Point Symmetry Groups w/ improper operations Any rotoinversion axis, except those for which n = 4N, can be decomposed into a combination of rotation axis & inversion or reflection

  8. Point Symmetry Groups not containing i G = g, ng = g, gn (g of index 2…cosets =…n transforms g into itself) For every pt grp G of 1st sort which can be expressed as product of subgroup g of index 2 and operation of 1st sort n (n even), can get a corresponding group G from same subgrp & 2nd sort operation n

  9. Point Symmetry Groups not containing i G = g, ng = g, gn (g of index 2…cosets =…n transforms g into itself) For every pt grp G of 1st sort which can be expressed as product of subgroup g of index 2 and operation of 1st sort n (n even), can get a corresponding group G from same subgrp & 2nd sort operation n 2 (=m) 4 6 (If n odd, order of n = 2x that of n)

  10. Point Symmetry

  11. Point Symmetry C4 C4 C4 C2 C2 C4 C2 C4 C2 C4 2 3 2 3 C4 C4 C4 m m C4 m C4 m C4 2 3 3 2

  12. Point Symmetry Groups containing i wrt i, any proper point group is of index 2 … i transforms any group into itself So, get new groups by G i

  13. Point Symmetry G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m

  14. Point Symmetry 4 i = 4/m C4 C2 C4 i i C4 i C2 i C4 G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m -1 -1

  15. Point Symmetry 4 i = 4/m C4 C2 C4 i i C4 i C2 i C4 2 i = 2/m C2 i i C2 G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m -1 -1

  16. Point Symmetry 4 i = 4/m C4 C2 C4 i i C4 i C2 i C4 2 i = 2/m C2 i i C2 2' i = 2'/m' C2 i i C2 G G i 1 1 2 2/m 3 3 4 4/m 6 6/m 222 2/m 2/m 2/m 32 3 2/m 422 4/m 2/m 2/m 622 6/m 2/m 2/m 23 2/m 3 432 4/m 3 2/m -1 -1 l l

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