Groups whose prime graph on class sizes has a cut vertex
Silvio Dolfi, Emanuele Pacifici, Luc\'ia Sanus, V\'ictor Sotomayor

TL;DR
This paper classifies finite groups based on the structure of their prime graph on class sizes, specifically identifying those groups whose prime graph has a cut vertex, revealing insights into their conjugacy class size distribution.
Contribution
It provides a complete classification of finite groups with a prime graph on class sizes that contains a cut vertex, a novel structural characterization.
Findings
Identifies all finite groups with a prime graph having a cut vertex.
Characterizes the structure of conjugacy class sizes in these groups.
Advances understanding of the relationship between group structure and prime graph properties.
Abstract
Let be a finite group, and let be the prime graph built on the set of conjugacy class sizes of : this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of , two vertices and being adjacent if and only if divides some conjugacy class size of . In the present paper, we classify the finite groups for which has a cut vertex.
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