| Summary: | <p>We construct a Lie relator which is not an identical Lie relator in the variety 𝔓<sub>4</sub>𝔓<sub>2</sub>. This is the first known example of a non-identical Lie relator.</p> <p>Next we consider the existence of torsion in outer commutator groups. Let <em>L</em> be a free Lie ring. Suppose that 1 < <em>i</em> ≤ <em>j</em> ≤ <em>2i</em> and <em>i</em> ≤ <em>k</em> ≤ <em>i</em> + <em>j</em> + 1. We prove that <em>L</em>/[<em>L<sup>j</sup></em>, <em>L<sup>i</sup></em>, <em>L<sup>k</sup></em>] is torsion free. Also, we prove that if 1 < <em>i</em> ≤ <em>j</em> ≤ 2<em>i</em> and <em>j</em> ≤ <em>k</em> ≤ <em>l</em> ≤ <em>i</em> + <em>j</em> then <em>L</em>/[<em>L<sup>j</sup></em>, <em>L<sup>i</sup></em>, <em>L<sup>k</sup></em>, <em>L<sup>l</sup></em></p>] is torsion free. We then prove that the analogous groups, namely <em>F</em>/[γ<sub><em>j</em></sub>(<em>F</em>),γ<sub><em>i</em></sub>(<em>F</em>),γ<sub><em>k</em></sub>(<em>F</em>)] and <em>F</em>/[γ<sub><em>j</em></sub>(<em>F</em>),γ<sub><em>i</em></sub>(<em>F</em>),γ<sub><em>k</em></sub>(<em>F</em>),γ<sub><em>l</em></sub>(<em>F</em>)] (under the same conditions for <em>i, j, k</em> and <em>i, j, k, l</em> respectively), are residually nilpotent and torsion free. We prove the existence of 2-torsion in the Lie rings <em>L</em>/[<em>L<sup>j</sup>, L<sup>i</sup>, L<sup>k</sup></em>] when 1 ≤ <em>k</em> < <em>i</em>,<em>j</em> ≤ 5, and thus show that our methods do not work in these cases. <p>Finally, we consider the order of finite groups of exponent 8. For <em>m</em> ≥ 2, we define the function <em>T</em>(<em>m</em>,<em>n</em>)</p> by <em>T(m</em>,1) = <em>m</em> and <em>T</em>(<em>m</em>,<em>k</em> + 1) = <em>m</em><sup><em>T</em>(<em>m</em>,<em>k</em>)</sup>. We prove that if <em>G</em> is a finite <em>m</em>-generator group of exponent 8 then |<em>G</em>| ≤ <em>T(m</em>, 7<sup>471</sup>), improving upon the best previously known bound of <em>T(m</em>, 8<sup>8<sup>8</sup></sup>).
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