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This quantity provides articles from 4 striking researchers who paintings on the cusp of research and good judgment. The emphasis is on energetic learn subject matters; many effects are offered that experience no longer been released sooner than and open difficulties are formulated. huge attempt has been made by means of the authors to make their articles obtainable to mathematicians new to the world

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**Example text**

2) T= U ImT(j). j E I ( A ,A ') Indeed, once this is done, it is clear that U ImT(j) = U U U ImT(flja) jEI(A,A') •EAut(A) flEAut(A') [j]EI(A,A') and it is easy to see that given i,jEI(A,A') and ctEAut(A), flEAut(A'), with i=fljot, then ImT(i)=T(c0(ImT(j)). 2). Given [r]E T, let A r be an NEC group with signature r' containing r(A) for some representative r of [r]. Let $:Ar-------~A' be an isomorphism and j =~br EI(A,A'). Clearly [r] = [~- l j] =Tfj)([q~" 1]) E ImT(j). Conversely, let j E I ( A , A ' ) , [s]ET(A') and [r]=T(j)([s])ET(A).

As in the first paragraph, the cases of odd or even N are studied separately. difference is that in case of even N, proper periods in F The main may proceed from period-cycles of F ' , whilst this is not the case for odd N. s. in short), which will be useful in the sequel. Definition 2. Let F E = { x 1 ..... X r } ~ F , be an NEC group. s, of F if the following conditions hold: is an 42 (i) Each elliptic element of /-" which is not a product of two reflections in F , is conjugate in /-" to a power of some x i E E .

Once this will be proved, a(/-') will consist of N/1i period-cycles P1 ..... PN/I i' are Pr: {1/2(order of the stabilizer in F of flreh(Nij))} whose elements 47 with j = 0 ..... s i and h = 0 ..... li-1. From the description (1) of the quotient F'/T, the map h-1 h t/ll. >(Brei) r/(Brei) is a bijection between StF(flre~(Nij))__ _ and StT,(Nij). Since the last order 2nij, we deduce that Ii Pr:(nil ..... his i ...... nil ..... his i) one has for each r = l ..... N/l i or with the periods in reverse order.