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Endomorphism and automorphism graphs of finite groups

Midhuna V Ajith, Peter J Cameron, Mainak Ghosh, Aparna Lakshmanan S

TL;DR

The paper defines and analyzes endomorphism and automorphism graphs associated with finite groups, embedding them in the framework of transformation monoids. It develops compressed and identity-deleted variants, studies their structural properties (planarity, girth, completeness, diconnectivity) and behavior under direct products, and provides detailed computations for cyclic groups, abelian $p$-groups, dihedral/dicyclic/symmetric/metacyclic groups. It also presents counterexamples to natural questions about isomorphisms of graphs and digraphs and discusses the relationship between endomorphism and power graphs. The results yield a nuanced portrait of how endomorphism structure encodes group-theoretic information, with complete classifications in several families and several open questions for nonabelian cases. The work contributes methodologically by leveraging the transformation-monoid viewpoint and conceptually by highlighting the limits of endomorphism data in distinguishing groups.

Abstract

Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping $a$ to $b$. The endomorphism graph, $\uend(G)$ is the corresponding undirected simple graph. The automorphism graph of $G$ is similarly defined for automorphisms: it is a disjoint union of complete graphs on the orbits of $\Aut(G)$. The endomorphism digraph is a special case of a digraph associated with a transformation monoid, and we begin by introducing this. We have explored graph theoretic properties like size, planarity, girth etc. and tried finding out for which types of groups these graphs are complete, diconnected, trees, bipartite and so on, as well as computing these graphs for some special groups. We conclude with examples showing that things are not always simple.

Endomorphism and automorphism graphs of finite groups

TL;DR

The paper defines and analyzes endomorphism and automorphism graphs associated with finite groups, embedding them in the framework of transformation monoids. It develops compressed and identity-deleted variants, studies their structural properties (planarity, girth, completeness, diconnectivity) and behavior under direct products, and provides detailed computations for cyclic groups, abelian -groups, dihedral/dicyclic/symmetric/metacyclic groups. It also presents counterexamples to natural questions about isomorphisms of graphs and digraphs and discusses the relationship between endomorphism and power graphs. The results yield a nuanced portrait of how endomorphism structure encodes group-theoretic information, with complete classifications in several families and several open questions for nonabelian cases. The work contributes methodologically by leveraging the transformation-monoid viewpoint and conceptually by highlighting the limits of endomorphism data in distinguishing groups.

Abstract

Let be a group. The directed endomorphism graph, of is a directed graph with vertex set and there is a directed edge from the vertex to the vertex if and there exists an endomorphism on mapping to . The endomorphism graph, is the corresponding undirected simple graph. The automorphism graph of is similarly defined for automorphisms: it is a disjoint union of complete graphs on the orbits of . The endomorphism digraph is a special case of a digraph associated with a transformation monoid, and we begin by introducing this. We have explored graph theoretic properties like size, planarity, girth etc. and tried finding out for which types of groups these graphs are complete, diconnected, trees, bipartite and so on, as well as computing these graphs for some special groups. We conclude with examples showing that things are not always simple.

Paper Structure

This paper contains 19 sections, 24 theorems, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Transformation monoids, preorders, and digraphs
  3. The endomorphism digraph of a group
  4. Direct products
  5. Cyclic groups
  6. Abelian Groups
  7. Abelian $\mathbf{p}$-groups
  8. $(\mathbb{Z}_{p^m})^l$ under addition $(+_{p^m}, +_{p^m}, \ldots +_{p^m})$
  9. $\mathbb{Z}_{{p}^{m_1}}^{l_1} \times \mathbb{Z}_{{p}^{m_2}}^{l_2} \times \ldots \times \mathbb{Z}_{{p}^{m_k}}^{l_k}$
  10. Identity element deleted subgraphs
  11. Compressed endomorphism digraph of dihedral, dicyclic, symmetric and metacyclic groups
  12. Dihedral Group
  13. Dicyclic group
  14. Symmetric group
  15. Some metacyclic groups
  16. ...and 4 more sections

Key Result

Proposition 4.2

Let $G$ and $H$ be groups with coprime orders. Then the endomorphism monoid of $G\times H$ is $\mathop{\mathrm{End}}(G)\times\mathop{\mathrm{End}}(H)$, and the endomorphism digraph of $G\times H$ is the strong product of $\mathop{\overrightarrow{\mathrm{EG}}}\nolimits(G)$ and $\mathop{\overrightarro

Figures (5)

  • Figure 1: Lattice diagram of $\mathbb{Z}_{12}$
  • Figure 2: Formation of a cycle in $\mathop{\mathrm{EG^*}}\nolimits$
  • Figure 3: Compressed endomorphism digraph of dihedral group for odd and even cases, $d$ dividing $k$
  • Figure 4: $\mathop{\overrightarrow{\mathrm{EG}}}\nolimits _-(Dic_n)$, when $n \equiv 2 (mod \, 4)$
  • Figure 5: Compressed endomorphism digraph of $S_6$ and $S_4$

Theorems & Definitions (49)

  • Definition 1.1
  • Proposition 4.2
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • proof
  • Theorem 5.4
  • proof
  • Example 5.5
  • Remark 5.6
  • ...and 39 more