Endomorphism Digraphs in Group Theory

• Endomorphism Digraph is a directed graph where each vertex represents a group element and an edge exists from a to b if an endomorphism maps a to b.
• It reveals a rich interplay between algebraic and graph-theoretic properties, highlighting characteristics like connectivity, completeness, and planarity.
• Applications span cyclic, abelian, and nonabelian groups, using the digraph structure to analyze transformations and invariant properties.

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 (Ajith et al., 2 Mar 2025, Ajith et al., 19 Nov 2025).

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 $\dend(G)$ (Ajith et al., 19 Nov 2025).

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 (Ajith et al., 2 Mar 2025).

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 (Ajith et al., 2 Mar 2025, Ajith et al., 19 Nov 2025).

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 (Ajith et al., 19 Nov 2025).

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 (Ajith et al., 2 Mar 2025).

• 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 $G \cong (\mathbb{Z}_p)^k$ (Ajith et al., 2 Mar 2025, Ajith et al., 19 Nov 2025).
• 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 (Ajith et al., 2 Mar 2025, Ajith et al., 19 Nov 2025).

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 $p$, $a\geq1$, $m,n\geq0$ (Ajith et al., 19 Nov 2025).

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 (Sha, 2010).

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 (Sha, 2010).

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 (Ajith et al., 19 Nov 2025).

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) (Ajith et al., 19 Nov 2025).

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 (Ajith et al., 19 Nov 2025).

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 (Ajith et al., 2 Mar 2025).
• Enumeration and Invariants: Exact formulas, sharp bounds, and reachability properties for arc count, diameter, or Hamiltonicity in nonabelian and product-group cases are unresolved (Ajith et al., 19 Nov 2025).
• 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 (Ajith et al., 19 Nov 2025).

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 (Sha, 2010). Further comprehensive surveys and classifications are found in (Ajith et al., 2 Mar 2025) and (Ajith et al., 19 Nov 2025).