Finite groups with some restriction on the vanishing set

TL;DR

This paper investigates finite groups where the gcd of the orders of vanishing elements in different conjugacy classes is at most 2, proving such groups are solvable and characterizing their structure when supersolvable.

Contribution

It establishes solvability for groups with restricted gcd conditions on vanishing elements and describes their structure in the supersolvable case.

Findings

Groups with gcd ≤ 2 for orders of vanishing elements are solvable.

Supersolvable groups with this property have a normal metabelian 2-complement.

The paper provides structural insights into these finite groups.

Abstract

Let $ x $ be an element of a finite group $ G $ and denote the order of $ x $ by $ \mathrm{ord}(x) $. We consider a finite group $ G $ such that $ \gcd(\mathrm{ord}(x),\mathrm{ord}(y))\leqslant 2 $ for any two vanishing elements $ x $ and $ y $ contained in distinct conjugacy classes. We show that such a group $ G $ is solvable. When $ G $ with the property above is supersolvable, we show that $ G $ has a normal metabelian $ 2 $-complement.