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 1st sort operations in such a group G form a
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 = a … an
Bg = B Ba … Ban
B is any 2nd sort operation in G

3. Point Symmetry 1st sort operations in such a group G form a
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 = a … an
Bg = B Ba … 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)
(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 2 3

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
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 -1 -1 G G i 1 1
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 -1 2 i = 2/m -1 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 -1 2 i = 2/m -1 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