chemia - Studia

TOPOLOGICAL SYMMETRY OF TWO FAMILIES OF DENDRIMERS
40
39
41
38
37
19
42
36
18
35
8
20
17
9
43
3
34
16
7
21
44
2
33
45
Figure 1. The Forth Generation of
Dendrimer Molecule D 1 [4].
6
15
1
4
10
32
22
14
5
31
11
23
30
13
29
12
24
25
28
26
27
49
50
47
48
46
51
45
22
23
52
21
24
44
9
10
53
43
54
42
20
3
25
41
19
8
2
18
7
1
4 5
11 12
26
55 56
40
27
39
17
38
6
37
36
16
13
15
14
35
28 29
60
Figure 2. The Forth Generation of
Dendrimer Molecule D 2 [4].
57
58
59
30
34
33
32
31
At first, we consider the dendrimer molecule D 1 [n], Figure 1. In order
to characterize the symmetry of this molecule we note that each dynamic
symmetry operation of D 1 [1], considering the rotations of XY 2 groups in
different generations of the whole molecule D 1 [n], is composed of n
sequential physical operations. We first have a physical symmetry of the
framework (as we have to map the XY 2 groups on XY 2 groups which are on
vertices of the framework). Such operations form the group G of order 6,
which as is well known to be isomorphic to S 3 or Sym(3). After accomplishing
the first framework symmetry operation we have to map each of the three
XY 2 group on itself in the first generation and so on. This is a group
isomorphic to H = ((…(Z 2 ∿ Z 2 ) ∿ Z 2 ) ∿ … )∿ Z 2 ) ∿ Z 2 with n – 1 components.
Therefore, the whole symmetry group is isomorphic to H ∿ G. This is a group
of order .
We now compute a generator set for this group. To do this, we apply
computer algebra system GAP to find a generating set for D 1 [2], D 1 [3] and
D 1 [4], see Table 1.
Table 1. Generating Sets for D 1 [2], D 1 [3] and D 1 [4].
a 1 = (1,2) a 2 = (1,2,3) b 1 = (4,5) b 2 = (6,7) b 3 = (8,9) b 4 = (10,11)
b 5 = (12,13) b 6 = (14,15) b 7 = (16,17) b 8 = (18,19) b 9 = (20,21) b 10 = (22,23)
b 11 = (24,25) b 12 = (26,27) b 13 = (28,29) b 14 = (30,31) b 15 = (32,33) b 16 = (34,35)
b 17 = (36,37) b 18 = (38,39) b 19 = (40,41) b 20 = (42,43) b 21 = (44,45)
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