Abstract:

Let $G\neq 1$ be a finite group and let $\mathbb{P}$ be the set of all primes. 
A chain $1=M_0 < M_1< \ldots  < M_{n-1}< M_n=G$ such that $M_i$ is a maximal subgroup of $M_{i+1}$ for every $i$ is called a maximal chain of~$G$. Every chain is associated with a sequence of non-negative integers $j_1$, $j_2$, \ldots, $j_n$, where $j_i=|M_i:M_{i-1}|$. A maximal chain is a $\mathbb{P}$-chain if $j_i\in\mathbb{P}$ for every $i$. We say that a $\mathbb{P}$-chain is a $\mathbb{P}^<$-chain ($\mathbb{P}^>$-chain) if $j_1\leq j_2\leq \ldots\leq j_n$ ($j_1\geq j_2\geq \ldots\geq j_n$, respectively). We investigate finite groups in which some maximal chains
are $\mathbb{P}$-chains. In particular, we obtain the following criteria for finite groups to be supersolvable:
a group $G$ is supersolvable if and only if there are a $\mathbb{P}^>$-chain and $\mathbb{P}^<$-chain in $G$;
a group $G$ is supersolvable if and only if $G$  has a Sylow tower of supersolvable type and
there is a $\mathbb{P}^<$-chain in $G$. The obtained results are used for characterization
of generally supersolvable groups.