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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…
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}…
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…
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.
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…
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…