[loops] 7 Specific methods

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# 7 Specific methods

### Sections

1. Moufang modifications
2. Triality for Moufang loops

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

## 7.1 Moufang modifications

Ale\vs Drápal discovered two prominent families of extensions of Moufang loops. These extensions can be used to obtain many, perhaps all, nonassociative Moufang loops of order at most 64. We call these two constructions Moufang modifications . The library of Moufang loops included with LOOPS is based on Moufang modifications. We describe the two modifications briefly here. See DV 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 ∈ S∩Z(L). Let α be a generator of L/S and write L = ∪i ∈ M αi, where M={−m+1, ..., m}. Let σ:Z→ M be defined by σ(i)=0 if i ∈ M, σ(i)=1 if i > m, and σ(i)=−1 if i < −m+1. Introduce a new multiplication ∗ on L defined by

x∗y = xyhσ(i+j),

where x ∈ αi, y ∈ αj, i ∈ M, j ∈ M. Then (L,∗) is a Moufang loop, a cyclic modification of L.

When L, S, α, h are as above and when a is any element of α, 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 ≥ 1. Let M and σ be defined as in the cyclic case. Let β, γ ∈ L/S be two involutions of L/S such that α = βγ generates a cyclic subgroup of L/S of order 2m. Let e ∈ β and f ∈ γ be arbitrary. Then L can be written as a disjoint union L=∪i ∈ M(αi∪eαi), and also L=∪i ∈ M(αi∪αif. Let G0=∪i ∈ Mαi, and G1=L\G0. Let h ∈ S∩N(L)∩Z(G0). Introduce a new multiplication ∗ on L defined by

  x∗y = xyh(−1)rσ(i+j),

  where x ∈ αi∪eαi, y ∈ αj∪αjf, i ∈ M, j ∈ M, y ∈ Gr, r ∈ {0,1}. Then (L,∗) is a Moufang loop, a dihedral 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 ∈ G} be a set of new elements. Define multiplication ∗ on L=G∪[(G)] by

  g∗h = gh, g∗ | h | = | hg | ,\ | g | ∗h = | gh−1 | ,\ | g | ∗ | h | =h−1g,

  where g, h ∈ 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`

  ## 7.2 Triality for Moufang loops

  Let G be a group and σ, ρ be automorphisms of G, satisfying σ2 = ρ3 = (σρ)2 = 1. We write the automorphisms of a group as exponents and [g,σ] for g−1gσ. We say that the triple (G,ρ,σ) is a group with triality if [g, σ] [g,σ]ρ [g,σ]ρ2 = 1 holds for all g ∈ G. It is known that one can associate a group with triality (G,ρ,σ) in a canonical way with a Moufang loop L. See NV 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 σ and ρ are in the same family as the elements of G.

  Given a Moufang loop L, the function

- `TrialityPermGroup(` L`) F`

  returns a record [G, ρ, σ], where G is the group with triality associated with L, and ρ, σ 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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