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math.GR2026

Some answers regarding factorizations of finite groups

Ryan McCulloch

In this note we answer some questions of G. M. Bergman concerning factorizations of finite groups. Such questions originated when M. H. Hooshmand asked whether, given a finite grou…

math.GR2026

A Möbius function on the centralizer lattice

William Cocke, Mark L. Lewis, Ryan McCulloch

We consider the Möbius function on the poset of element centers and obtain some new results regarding centralizers in a -group.

math.GR2026

An answer regarding automorphisms of finite abelian groups

Ryan McCulloch

In this note we provide a negative answer to the question: ``Is it true that for every positive rational number there exists a finite abelian group such that $|\mathrm{Aut}…

math.GR2025

Covering by Centralizers

Mark L. Lewis, Ryan McCulloch

In this paper, we consider covers of finite groups by centralizers of elements. We show that the set of centralizers that are maximal under the partial ordering form a cover of the…

math.GR2025

The commuting graph and a graph associated with centralizers

Mark L. Lewis, Ryan McCulloch

Let be a -group. We begin to consider the relationship between the structure of the commuting graph and . We also build a family of groups whose commuting graphs h…

math.GR2025

Finite groups with dense -subgroups

Ryan McCulloch, Marius Tărnăuceanu

A group is said to have dense -subgroups if each non-empty open interval of the subgroup lattice contains a subgroup in the Chermak--Delgado lattice ${\cal CD…