On the factorization graph of finite groups
Abstract
Given a finite group G, the factorization graph Γ(G) has as vertices the nontrivial proper subgroups of G, with two distinct vertices H and K adjacent if and only if HK = G. In this paper, we study structural and combinatorial properties of Γ(G) for three families of groups: cyclic groups of square-free order, generalized quaternion groups, and dicyclic groups. For cyclic groups Cp1⋯ pr, we describe the global structure of Γ(G) and determine its order, degree sequence, center, girth, and diameter. We also characterize completely when Γ(G) is claw-free, show that it is non-planar for r ≥ 4, and prove that after removing all pendant vertices, the resulting induced subgraph is 2-connected; we further compute its exact circumference for small r, showing it is Hamiltonian precisely when r=4, 5. For generalized quaternion groups, we classify all isolated vertices and prove that the reduced factorization graph has diameter two and a unique central vertex. Moreover, we show that it is planar, has chromatic number~$3$, is Eulerian, and non-Hamiltonian, and that it contains maximal induced stars and triangular book subgraphs. Finally, we analyze the dicyclic case and show that the reduced factorization graph is a complete bipartite graph, from which all main invariants follow directly. An explicit SageMath algorithm for constructing Γ(G) is included.
Related
- On the Generating Graph of Finite Abelian Groups
- Lambda number of the reduced power graph of finite groups
- Generalized Graph Compositions with Applications to Difference Graphs of Finite Groups
- Torsion of every finite order in the homology of graph braid groups
- Cyclotomic factorization and fixed-point insertion in colored permutation groups
BibTeX
@misc{gutierrez2026factorization,
title = {{On the factorization graph of finite groups}},
author = {Ismael Gutiérrez and Yesneri Zuleta y E. J. García-Claro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40860},
eprint = {2610.40860},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40860},
}
RIS
TY - UNPB
TI - On the factorization graph of finite groups
AU - Ismael Gutiérrez
AU - Yesneri Zuleta y E. J. García-Claro
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40860
UR - https://arxiv.org/abs/2610.40860
AB - Given a finite group G, the factorization graph Γ(G) has as vertices the nontrivial proper subgroups of G, with two distinct vertices H and K adjacent if and only if HK = G. In this paper, we study structural and combinatorial properties of Γ(G) for three families of groups: cyclic groups of square-free order, generalized quaternion groups, and dicyclic groups. For cyclic groups C_p_1⋯ p_r, we describe the global structure of Γ(G) and determine its order, degree sequence, center, girth, and diameter. We also characterize completely when Γ(G) is claw-free, show that it is non-planar for r ≥ 4, and prove that after removing all pendant vertices, the resulting induced subgraph is 2-connected; we further compute its exact circumference for small r, showing it is Hamiltonian precisely when r=4, 5. For generalized quaternion groups, we classify all isolated vertices and prove that the reduced factorization graph has diameter two and a unique central vertex. Moreover, we show that it is planar, has chromatic number~$3$, is Eulerian, and non-Hamiltonian, and that it contains maximal induced stars and triangular book subgraphs. Finally, we analyze the dicyclic case and show that the reduced factorization graph is a complete bipartite graph, from which all main invariants follow directly. An explicit SageMath algorithm for constructing Γ(G) is included.
ER -