Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Characterizations of some classes of finite $\sigma$-soluble $P\sigma
  T$-groups   Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all
primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if
every chief factor $H/K$ of $G$ is a $\sigma _{i}$-group for some $i=i(H/K)$.
  A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma
$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma
_{i}$-subgroup of $G$ for some $\sigma _{i}\in \sigma $ and ${\cal H}$ contains
exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i \in I$ such that
$\sigma _{i}\cap \pi (G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be
${\sigma}$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set $\cal H$
such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \cal H$.
  We obtain characterizations of finite $\sigma$-soluble groups $G$ in which
$\sigma$-permutability is a transitive relation in $G$.

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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th