On $\sigma$-properties of finite groups I

Let $\sigma=\{\sigma_i|i \in I\}$ be some partition of the set $\mathbb{P}$ of all primes, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$. We say that a finite group $G$ is: $\sigma$-primary if $G$ is a $\sigma_i$-group for some $\sigma_i\in\sigma$; a $\sigma$-group if $G$ has a set $\mathcal{H}=\{H_1, \dots, H_t\}$ of Hall subgroups such that $H_i$ is $\sigma$-primary, $(|H_i|, |H_j|)=1$ for all $i\ne j$ and $\pi(G)=\pi(H_1)\cup\dots\cup\pi(H_t)$. We analyze some properties of finite $\sigma$-groups.