---
title: Endomorphism Digraphs in Group Theory
url: https://www.emergentmind.com/topics/endomorphism-digraph
type: topic
---

# Endomorphism Digraphs in Group Theory

An endomorphism digraph of a group $G$ is a simple directed graph whose vertices correspond to the elements of $G$ and in which there is a directed edge from $a$ to $b$ $(a \neq b)$ if and only if there exists an endomorphism $\varphi\in\mathrm{End}(G)$ with $\varphi(a)=b$. This construction encodes the action of the endomorphism monoid on $G$ into combinatorial structure, and it gives rise to a rich interaction between algebraic group properties and structural graph invariants. Endomorphism digraphs are intrinsically linked to the study of transformation monoids and the interaction between group theory and graph theory. 

## 1. Formal Definition and Fundamental Properties

Given a group $G$, the directed endomorphism graph — denoted $\overrightarrow{\mathrm{Endo}}(G)$ or, in shorthand, $\dend(G)$ — is a simple digraph defined by:
- Vertex set $V(\dend(G)) = G$.
- For $a, b \in G$, a directed arc $a \to b$ (where $a\neq b$) exists if and only if $\exists\,\varphi \in \mathrm{End}(G)$ such that $\varphi(a) = b$.
- No loops (no $(a\to a)$) and no multiple edges.

Formally,
$$
V(\dend(G)) = G, \qquad
E(\dend(G)) = \{ (a, b) \in G \times G \mid a \neq b,\, \exists\, \varphi \in \mathrm{End}(G) : \varphi(a) = b \}.
$$
The undirected endomorphism graph $\Gamma_{\mathrm{End}}(G)$ is formed by suppressing orientations and multiplicities. The corresponding automorphism graph of $G$ considers automorphisms in place of endomorphisms and typically manifests as a disjoint union of complete graphs on $\mathrm{Aut}(G)$-orbits [2503.00759, 2511.15602].

The relation $\exists\,\varphi\in\mathrm{End}(G)\!: \varphi(a)=b$ yields a reflexive (if loops included) and transitive relation, so the preorder induced on $G$ by endomorphisms can be fully represented by the structure of $\dend(G)$ [2511.15602].

## 2. Illustrative Examples and Special Cases

### Cyclic Groups ($\mathbb{Z}_n$)
For $G = \mathbb{Z}_n$, every endomorphism is “multiply by $k$” ($k=0,\ldots,n-1$), and
$$
x \to y \iff y \in \langle x \rangle,
$$
making $\dend(\mathbb{Z}_n)$ isomorphic to the directed power graph. In particular, for $n=p$ prime, every nonzero $x$ generates $G$ and each $x\neq0$ has arcs to all $y\neq x$, and to $0$ via the trivial map, yielding the complete digraph on $p$ vertices (excluding loops). For $G\cong(\mathbb{Z}_p)^k$, the digraph is complete on $p^k$ vertices [2503.00759].

In $\mathbb{Z}_6$, arcs $a \to b$ occur precisely when $\mathrm{ord}(b) \mid \mathrm{ord}(a)$, reflecting the divisibility structure among orders.

### Nonabelian Simple Groups
Here, $\mathrm{End}(G)=\{\text{trivial map}\}\cup\mathrm{Aut}(G)$. The only nontrivial endomorphisms are automorphisms, with the trivial map sending every $a\neq e$ to $e$. The digraph is a disjoint union comprising sinks at $e$ (from trivial map) and clusters corresponding to $\mathrm{Aut}(G)$-orbits [2503.00759, 2511.15602].

### Structured Families (Abelian, Dihedral, Dicyclic, Symmetric Groups)
- For abelian $G$, arcs $a\to b$ exist iff $|b|$ divides $|a|$. If all nonzero elements have the same prime order, the non-identity subgraph is complete (elementary abelian case).
- Dihedral $D_{2n}$: The compressed digraph on rotations is isomorphic to the compressed $\dend(\mathbb{Z}_n)$; for $n$ even, additional arcs between rotation- and reflection-classes occur.
- Dicyclic and metacyclic groups: Similar divisibility structures with extra vertices corresponding to special cosets or reflections arise.
- Symmetric $S_n$: The only nontrivial normal subgroup is $A_n$ (except $n=6$), so arcs correspond to the unique nonzero endomorphism $S_n \to \mathbb{Z}_2$ factoring through $S_n/A_n$ and inner automorphisms [2511.15602].

## 3. Graph Invariants and Characterizations

Several classical graph-theoretic invariants exhibit group-theoretic fingerprints:

- **Size**: For cyclic $G=\mathbb{Z}_n$, the total number of arcs is
$$
\sum_{d\mid n} \phi(d) (\phi(d)-1)
$$
where $\phi$ is Euler's totient [2503.00759].
- **Strong connectivity** is absent in $\dend(G)$ because the identity $e$ has no outgoing arcs. However, upon deleting $e$, the following are equivalent for $\dend^*=\dend|_{G\setminus\{e\}}$: (1) strong connectivity, (2) completeness, (3) existence of a Hamiltonian cycle. For abelian $G$, this occurs if and only if $G \cong (\mathbb{Z}_p)^k$ [2503.00759, 2511.15602].
- **Girth**: For $|G|>2$, the girth of $\Gamma_{\mathrm{End}}(G)$ is 3, due to triangles formed by any non-trivial automorphism.
- **Bipartiteness**: Only possible for $G=\mathbb{Z}_2$.
- **Planarity**: For abelian groups, planarity is restricted to $|G|\leq 4$, i.e., $\mathbb{Z}_2$, $(\mathbb{Z}_2)^2$, $\mathbb{Z}_3$, and $\mathbb{Z}_4$; otherwise, the presence of an induced $K_5$ or $K_{3,3}$ is unavoidable [2503.00759, 2511.15602].

A table succinctly summarizes these results for abelian groups:

| Invariant         | Characterization                                      | Extreme Cases       |
|-------------------|------------------------------------------------------|---------------------|
| Strong Connectivity ($\dend^*$) | $G \cong (\mathbb{Z}_p)^k$                           | Complete digraph    |
| Planarity         | Only if $|G|\leq 4$                                  | $K_2, K_3, K_4$     |
| Bipartiteness     | $G\cong \mathbb{Z}_2$                                | Yes                 |
| Girth             | $\geq 3$ if $|G|>2$                                  | $G=\mathbb{Z}_2$: N/A|

Complete digraphs among undirected endomorphism graphs occur if and only if $G\cong (\mathbb{Z}_{p^a})^m \times (\mathbb{Z}_{p^{a+1}})^n$ for some $p$, $a\geq1$, $m,n\geq0$ [2511.15602].

## 4. Endomorphism Digraphs of Cyclic Groups as Dynamical Objects

For $G=\langle g \rangle$, $|G|=n$, and endomorphism $\varphi_k:g\mapsto g^k$, the endomorphism digraph $D(\varphi_k,G)$ has:
- Every vertex out-degree 1.
- In-degree either $d = \gcd(n,k)$ or $0$.
- Each component consists of exactly one directed cycle (elements of order dividing $t$, where $n = tk$ and $(t, k)=1$), with attached rooted trees (arborescences) of bounded height.
- The adjacency matrix is a permutation matrix or, in more generality, a single $1$ per row, and its spectral properties reveal cycle and tree structure [1007.1712].

The automorphism group of $D(\varphi_k,G)$ decomposes as a wreath product of the automorphism groups of the isomorphism classes of its connected components [1007.1712].

## 5. Connections to Group Structure: Compression, Products, and Counterexamples

**Compression**: Contracting automorphism-orbits or deleting the identity yields quotient graphs reflecting divisor posets or Hasse diagrams of order lattices, especially for abelian and cyclic groups.

**Direct Product Behavior**: For coprime groups $G$, $H$, $\dend(G\times H) \cong \dend(G)\boxtimes\dend(H)$ (strong product), so structurally unrelated groups may yield isomorphic endomorphism digraphs [2511.15602].

**Non-isomorphic Groups with Isomorphic Digraphs**: Three non-isomorphic groups of order $p^3$ ($\mathbb{Z}_{p^3},\ \mathbb{Z}_{p^2}\times\mathbb{Z}_p,$ and a nonabelian $p$-group) share the same preorder structure on element orders, and thus isomorphic endomorphism digraphs. Similarly, groups with isomorphic undirected endo-graphs may have non-isomorphic digraphs (e.g., nonabelian $p$-group of exponent $p$ vs. elementary abelian $p$-group) [2511.15602].

Automorphism graphs do not in general determine endomorphism digraphs: for some nonabelian groups, all $r$-elements fall into one endomorphism-class but split into several automorphism-classes [2511.15602].

## 6. Open Problems and Directions

- **Graph–Group Classification**: It remains open whether isomorphism of endomorphism digraphs implies group isomorphism; the converse to the fundamental theorem is conjectured for the directed (not undirected) case [2503.00759].
- **Enumeration and Invariants**: Exact formulas, sharp bounds, and reachability properties for arc count, diameter, or Hamiltonicity in nonabelian and product-group cases are unresolved [2511.15602].
- **Nonabelian Connectivities**: Full classification of nonabelian groups according to the strong connectivity or completeness of their endomorphism digraphs remains open.
- **Combinatorial–Algebraic Bridging**: Understanding how group decompositions or extensions control digraph invariants is an area of active study.

A plausible implication is that endomorphism digraphs provide a combinatorial invariant with substantial—but not complete—power to distinguish group structure, particularly when combined with compression or other quotient constructions.

## 7. Illustrative Models and Explicit Constructions

Concrete diagrams described in the literature highlight the intricacy:
- $\dend(\mathbb{Z}_4)$: Four vertices, with trivial-map arcs to $0$ and additional arcs from the inversion automorphism; undirected, it is $K_4$.
- $\dend(\mathbb{Z}_6)$ (compressed): Vertices correspond to divisors, arcs reflect order-divisibility.
- $\dend(D_8)$ (compressed): Vertices for $[r],[r^2],[s]$, with both $K_3$ structure and additional 2-cycles [2511.15602].

The interplay between combinatorial structure (order, orbits, cycle trees) and algebraic characteristics (endomorphism monoid, automorphism classes, normal subgroups) exemplifies the depth and variety encountered in the study of endomorphism digraphs.

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For further details, including explicit adjacency matrices, minimal polynomials, and automorphism group decompositions for endomorphism digraphs of cyclic groups, see [1007.1712]. Further comprehensive surveys and classifications are found in [2503.00759] and [2511.15602].

Source: https://www.emergentmind.com/topics/endomorphism-digraph