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Let $G$ be a finite group and $S< G$. A cover for a group $G$ is a collectionof subgroups of $G$ whose union is $G$. We use the term $n$-cover for a coverwith $n$ members. A cover $\Pi =\{H_1, H_2, \dots, H_n\}$ is said to be astrict $\mathit{S}$-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 (equallypartition, resp.).

Criteria for nilpotency of groups via partitons