Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Almost Engel finite and profinite groups   Let $g$ be an element of a group $G$. For a positive integer $n$, let
$E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$
over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a
profinite group such that for every $g\in G$ there is $n=n(g)$ such that
$E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$
is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that
Engel profinite groups are locally nilpotent. In the case of a finite group
$G$, we prove that if, for some $n$, $|E_n(g)|\leq m$ for all $g\in G$, then
the order of the nilpotent residual $\gamma _{\infty}(G)$ is bounded in terms
of $m$.

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The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
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The categories of this paper are: dg-ga