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# 8 Specific methods

### Sections

1. Core methods for Bol loops
2. Moufang modifications
3. Triality for Moufang loops

This chapter describes methods of LOOPS that apply to some special loops, mostly Bol and Moufang loops.

## 8.1 Core methods for Bol loops

Let L be a left Bol loopindexleft Bol loop such that the mapping x® x2 is a permutation of L. Define a new operation * on L by x*y = (x(y2x))1/2. Then Bruck showed that (L,*) is a left Bruck loop, called the assoicated left Bruck loop.indexassociated left Bruck loop. (In fact, Bruck used the isomorphic operation x*y = x1/2(yx1/2) instead. Our approach is more natural, since the associated left Bruck loop to a left Bruck loop L is identical to L then.) The associated left Bruck loop of L is returned via

- `AssoicatedLeftBruckLoop(` L`) A`

  ## 8.2 Moufang modifications

  Ale\vs Drápal discovered two prominent families of extensions of Moufang loops. It turns out that these extensions can be used to obtain all nonassociative Moufang loops of order at most 64. We call these two constructions Moufang modificationsindexMoufang modifications. The library of Moufang loops included with LOOPS is based on Moufang modifications. We describe the two modifications briefly here. See DrVo for details.

  Assume that L is a Moufang loop with normal subloop S such that L/S is a cyclic group of order 2m. Let h Î S Ç Z(L). Let a be a generator of L/S and write L = Èi Î M ai, where M={-m+1, ..., m}. Let s:Z® M be defined by s(i)=0 if i Î M, s(i)=1 if i > m, and s(i)=-1 if i < -m+1. Introduce a new multiplication * on L defined by

  x*y = xyhs(i+j),

  where x Î ai, y Î aj, i Î M, j Î M. Then (L,*) is a Moufang loop, a cyclic modificationindexcyclic modification of L.

  When L, S, a, h are as above and when a is any element of a, the corresponding cyclic modification is obtained via

- `LoopByCyclicModification(` L`,` S`,` a`,` h`) F`

  Now assume that L is a Moufang loop with normal subloop S such that L/S is a dihedral group of order 4m, with m ³ 1. Let M and s be defined as in the cyclic case. Let b, g Î L/S be two involutions of L/S such that a = b g generates a cyclic subgroup of L/S of order 2m. Let e Î b and f Î g be arbitrary. Then L can be written as a disjoint union L=Èi Î M(aiÈ e ai), and also L=Èi Î M(aiÈ aif. Let G0=Èi Î Mai, and G1=L\G0. Let h Î S Ç N(L)Ç Z(G0). Introduce a new multiplication * on L defined by

  x*y = xyh(-1)rs(i+j),

  where x Î aiÈ e ai, y Î ajÈ ajf, i Î M, j Î M, y Î Gr, r Î {0,1}. Then (L,*) is a Moufang loop, a dihedral modificationindexdihedral modification of L.

  When L, S, e, f and h are as above, the corresponding dihedral modification is obtained via

- `LoopByDihedralModification(` L`,` S`,` e`,` f`,` h`) F`

  In order to apply the cyclic and dihedral modifications, it is beneficial to have access to a class of nonassociative Moufang loops. The following construction is due to Chein:

  Let G be a group. Let [`(G)]={[`(g)]; g Î G} be a set of new elements. Define multiplication * on L=G È[`(G)] by

  g*h = gh, g*

  h

  =

  hg

  ,\

  g

  *h =

  gh-1

  ,\

  g

  *

  h

  =h-1g,

  where g, h Î G. Then L=M(G,2) is a Moufang loop that is nonassociative if and only if G is nonabelian.

  The loop M(G,2) can be obtained from a finite group G with

- `LoopMG2(` G`) F`

  ## 8.3 Triality for Moufang loops

  Let G be a group and s, r be automorphisms of G, satisfying s2 = r3 = (s r)2 = 1. We write the automorphisms of a group as exponents and [g,s] for g-1gs. We say that the triple (G,r,s) is a group with trialityindexgroup with triality if [g, s] [g,s]r [g,s]r2 = 1 holds for all g Î G. It is known that one can associate a group with triality (G,r,s) in a canonical way with a Moufang loop L. See NaVo2003 for more details.

  For any Moufang loop L, we can calculate the triality group as a permutation group acting on 3|L| points. If the multiplication group of L is polycyclic, then we can also represent the triality group as a pc group. In both cases, the automorphisms s and r are in the same family as the elements of G.

  Given a Moufang loop L, the function

- `TrialityPermGroup(` L`) F`

  returns a record [G, r, s], where G is the group with triality associated with L, and r, s are the corresponding triality automorphisms.

  The function

- `TrialityPcGroup(` L`) F`

  differs from `TrialityPermGroup` only in that G is returned as a pc group.

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