Abstract Algebra — Descending Endomorphisms

New constructs for mappings between group and graph theory.

Investigated a new definition for certain types of endomorphisms of groups called “descending” endomorphisms.
Implemented programs in SageMath to compute descending endomorphisms of groups for proof validation.
Analyzed representations of descending endomorphisms in graph theory.

A descending endomorphism $\delta$ of a group $G$ is one that descends to every quotient: for each normal subgroup $N \trianglelefteq G$, the quotient map $\varphi : G \to G/N$ satisfies $\varphi \circ \delta = \bar{\delta} \circ \varphi$ for a well-defined induced map $\bar{\delta}$ on $G/N$. These maps can be represented as directed graphs on the group’s elements, connecting group and graph theory.

Descending endomorphism graph of the dicyclic group Dic₄

Publications

Vinay Madhusudanan, Arjit Seth, and G. Sudhakara. “Descending Endomorphisms of Groups”. Palestine Journal of Mathematics 12.1 (2023), pp. 318–325.
Vinay Madhusudanan, Arjit Seth, and G. Sudhakara. “Descending endomorphisms of some families of groups”. In: Applied Linear Algebra, Probability and Statistics: A Volume in Honour of C. R. Rao and Arbind K. Lal. Springer, 2023, pp. 409–424.
Vinay Madhusudanan, G. Sudhakara, and Arjit Seth. “Descending endomorphism graphs of groups”. AKCE International Journal of Graphs and Combinatorics 20.2 (2023), pp. 148–155. doi:10.1080/09728600.2023.2234956