On diameters estimations of the commuting graphs of Sylow $p$-subgroups of the symmetric groups

Abstract

The commuting graph of a group $G$ is an undirected graph whose vertices are non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if and only if $xy=yx$. This article deals with the properties of the commuting graphs of Sylow $p$-subgroups of the symmetric groups. We define conditions of connectedness of respective graphs and give estimations of the diameters if graph is connected.

Keywords

• commuting graph
• wreath product
• sylow $p$-subgroup
• symmetric group