On F-groups with the central factor of order p 4

Abstract

Abstract A finite group G is called an F-group (G ∈ F) if for every x, y ∈ G ∖ Z(G), C G (x) ≤ C G (y) implies that C G (x) = C G (y). An important subclass of F-groups are CA-groups, consisting of groups in which all centralizers of noncentral elements are abelian. In this paper, among other results, we find the number of element centralizers and the maximum cardinality of subsets of pairwise non-commuting elements in an F-group G with | G Z ( G ) $\frac{G}{Z(G)}$ | = p 4 for some prime p.