[GAP Forum] Non-left n-Engel elements in a group

[Stefan Kohl]

On Thu, September 19, 2013 8:05 pm, yassine Guerboussa wrote:
> I will be pleased if one can help in writing a program such that,
> For a group $G$ (usually, a small group in GAP library), it computes the subgroup
> generatedby the elements that are not left n-Engel, for some fixed n.
> Recall that an element $a$ in $G$ is left n-Engel if it satisfies the
> identity$[x,a,...,a]=1$, where $a$ occurs $n$ times (the commutator is left normed).

I think the following should serve the purpose, as long as your groups are small:

SubgroupGeneratedByElementsWhichAreNotLeftnEngel := function ( G, n )

  local  elms, gens, H;

  elms := AsList(G);
  gens := Filtered(elms,
            a->ForAny(elms,x->not IsOne(LeftNormedComm(
                 Concatenation([x],ListWithIdenticalEntries(n,a))))));
  H := Group(gens,One(G));
  return H;
end;

Hope this helps,

    Stefan Kohl

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