On The Critical Number Of Finite Groups (II)

Abstract

Let $G$ be a finite group and $S\subseteq G\setminus \{0\}$. We call $S$ an additive basis of $G$ if every element of $G$ can be expressed as a sum over a nonempty subset in some order. Let $cr(G)$ be the smallest integer $t$ such that every subset of $G\setminus \{0\}$ of cardinality $t$ is an additive basis of $G$. In this paper, we determine $cr(G)$ for the following cases: (i) $G$ is a finite nilpotent group; (ii) $G$ is a group of even order which possesses a subgroup of index $2$.