By Hendrik Moraal (eds.)

ISBN-10: 354013896X

ISBN-13: 9783540138969

ISBN-10: 3540391088

ISBN-13: 9783540391081

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**Example text**

Then the a u t o m o r p h i s m group holds, the graph is is imprimitive; but since 6~(G) ¶ n o n p e r m i s s i b l e p e r m u t a t i o n group. Then (im) primitive. have the same MI by definition, this follows ¶ As a first a p p l i c a t i o n of these lemmas, the nature of all p e r m i s s i - ble groups with two graphs in their MI's is clarified: T h e o r e m I. A p e r m i s s i b l e group 6 on M letters w i t h two graphs in its MI is either p r i m i t i v e and the a u t o m o r p h i s m group of one of the graphs o r it is a w r e a t h p r o d u c t S ( M I ) ~ S ( M 2) with M = M I M 2.

Set such that cc'c=x; m u l t i p l y i n g c'dsC; set this with c=ddl, c' on the left yields (c'd) d I (c'd) d1= d I 2 = i = c'x. c'cc'c = c'ddlc'dd I = Since c'2=i by (i), x=c' F r o m the above, C'={clceC, c follows. the sets proper}, C' and Tg(1)=g(1) for T Cld' C and c2d" gEC'; belong set Tg(1) =g -I (I) Clearly then, if the in step T to D and T is not equal to the abelian, (39) ~ gsD' , But now, the proper elem- clc 2 is such that for all d',d"~D. if Actually, if this group c2=d2d3 ; then one has c 2 c 1 = d 2 d 3 d l d 2 = C 2 d l C 2 d 3 - 1 = d l d 3 -I .

45 If 6~6' is completely permissible, itive iff (a) 6 is primitive, then also (iii) The q-fold direct product is normal in Proof. If 6 6. (ii) 6~6' (b) 6~C(p) and (c) is completely permissible, is prim- 6' is transitive. 6~6'. then also the q-fold direct product of this group with itself, so that (6~G') (p) is completely per- missible as an extension of the q-fold direct product. ®6. This implies that 6 is completely permissible, so that part (i) of the theorem is proved. (ii). The letter-l-fixing subgroup of variant.

### Classical, Discrete Spin Models: Symmetry, Duality and Renormalization by Hendrik Moraal (eds.)

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