Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Probabilistically nilpotent groups of class two   For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y,
z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite
groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq
\epsilon > 0$, $G$ has a class-4 nilpotent normal subgroup $H$ such that $[G :
H] $ and $|\gamma_4(H)|$ are both bounded in terms of $\epsilon$. We also show
that if $G$ is an infinite group whose commutators have boundedly many
conjugates, or indeed if $G$ satisfies a certain more general commutator
covering condition, then $G$ is finite-by-class-3-nilpotent-by-finite.

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The categories of this paper are: eess
30
The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
1
The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: stat