Centralizers and the maximum size of the pairwise noncommuting elements in finite groups

Abstract

In this article, we determine the structure of all nonabelian groups $G$ such that $G$ has the minimum number of the element centralizers amongnonabelian groups of the same order. As an application of this result, we obtain the sharp lower bound for $\omega(G)$ in terms of the order of $G$ where $\omega(G)$ is the maximum size of a set of the pairwise noncommuting elements of $G$.