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**Extra info for Classical, Discrete Spin Models: Symmetry, Duality and Renormalization**

**Example text**

IQ (p) i= 64 = s l Q [ , (20) (18) implies for the general case: (QoK(2 q))(p) = (S(2)~K(4))eK(2 q). This completes step (a). (21) 53 Step in eq. (b). Let R now be a regular group with an automorphism (4). The permutation Tg (I) =g -+1 (I) iff T ~ as defined by ~ (g) = g -+1 (22) is such that TgT = ~(g) holds; (23) this permutation keeps all graphs of the MI of R invariant: TGk=T{g (I) ,ggk(1) }={e(g) (I) ,~(ggk ) (I) }={~(g) (I) ,~(g)~(gk ) (I) }= ={g(1) ,g~(gk) (I) }={g(1),gg k NOW a subgroup T T = {glgsR, s~T; ±I (24) (I) }=Gk={g(1),ggk(1)}.

If case, so that the other graphs Therefore, A permissible is a repeated (i) of the lemma) In the latter graphs, M I letters. ~I permissible There that there are > qM/2. This shows on Now assume (9) . One then has the inequality M-I with so that it has (11) to each vertex. (say the first gives M=MkIMk2, = M-Mkl edges adjacent M-I graphs with (I 4) the lemma. 4. 3. Direct products. A further useful product of groups i~ the direct product transitive permutation groups, Definition I. Let 61 and and S2, respectively.

62, so that eq. I shows the MI of the $(2)0S(4), S(2)0$(4) (4) follows. groups corr- ¶ (completely permissible, see compare also Fig. 1. It is not difficult to is actually the group of the connected graph in this this connected graph is the three-dimensional cube. It is also 32 interesting graphs to note that the other graphs G (2) and G (3) of this cube G in Fig. 2. KCSI=I I "4_ i / z2=3 z1=3 Fig. I. The MI of the group In the following, in more detail. s 2 graphs 1 1 the number To this end, of the three-dimensional of graphs it is assumed of sets, different respectively.