Criteria for nilpotency of groups via partitons

Abstract

Let $G$ be a finite group and $S< G$. A cover for a group $G$ is a collection of subgroups of $G$ whose union is $G$. We use the term $n$-cover for a cover with $n$ members. A cover $\Pi =\{H_1, H_2, \dots, H_n\}$ is said to be a strict $\mathit{S}$-partition of $G$ if $H_i\cap H_j= S$ for $i\neq j$ and $\Pi$ is said an equal strict $\mathit{S}$-partition (or $ES$-partition ) of $G$, if $\Pi$ is a strict $S$-partition and $|H_i|=|H_j|$ for all $i\neq j$. If $S$ is the identity subgroup and $G$ has a strict $S$-partition (equal strict $\mathit{S}$-partition), then we say that $G$ has a partition (equally partition, resp.).