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September 5, 2025

Co-maximal Subgroup Graphs of Finite Groups

Key Points

The study identifies conditions for a co-maximal subgroup graph to be connected, particularly regarding normal subgroups.
In finite abelian groups, the diameter of the co-maximal subgroup graph is shown to be at most 3 and its structure is characterized.
Characterizations are provided for graphs where the graph of co-maximal subgroups forms complete bipartite graphs or null graphs.
For the semidirect product of finite cyclic groups, it is demonstrated that the corresponding graph is connected.

Abstract

For a finite group Formula: see text, the co-maximal subgroup graph Formula: see text of Formula: see text is a graph whose vertices are proper subgroups of Formula: see text, and two distinct vertices Formula: see text and Formula: see text are adjacent if and only if Formula: see text. The deleted co-maximal subgroup graph Formula: see text is obtained by removing isolated vertices from Formula: see text. Firstly, we provide necessary and sufficient conditions for Formula: see text to be connected; in particular, from the viewpoint of normal subgroups in Formula: see text, we give some sufficient conditions for Formula: see text to be connected. Secondly, for a finite abelian group Formula: see text we prove that the diameter of Formula: see text, Formula: see text, is at most 3. Also, we characterize Formula: see text with Formula: see text for Formula: see text and we give characterizations for Formula: see text with Formula: see text being complete bipartite graphs and null graphs separately. Finally, we show that for the semidirect product Formula: see text of two finite cyclic groups, Formula: see text is connected and Formula: see text.

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