Group Petri Nets: Structural and Algebraic Models for Finite Groups and Their Subgroups
Abstract
Every finite group (G, * ) determines a Petri net N G whose places are the elements of G and whose transitions realise the group's Cayley table: transition t a,b consumes one token from a and one from b and deposits one token in a * b. We develop this correspondence as a single connected story. Restricting N G to a subgroup H ≤ G gives a Subgroup Petri Net; separating a group element's carrier and control roles gives a control-copy net M G ; and restricting the control layer of M G to H makes one-step reachability on the carrier layer coincide exactly with the rightcoset partition of G by H, giving a Petri-net reading of Lagrange's theorem, |G| = [G : H] • |H|. We show every Group Petri Net carries a strictly decreasing token-sum functional, forcing boundedness and termination of every firing sequence, with the terminal marking independent of firing order for every initial marking exactly when G is abelian. All constructions are illustrated with fully worked, diagrammed examples: Z 3 for the basic construction, and S 3 , with the explicit subgroup H = {e, (1 2)} and its three coset blocks, for the subgroup, control-copy, and non-confluence phenomena.
Related
- Finite abelian subgroups of algebraic groups
- Classes of Subgroups of a finite group with isomorphic quotient groups
- On second maximal subgroups of finite groups
- Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups
- Finite groups whose proper subgroups of order divisible by p are supersolvable
BibTeX
@misc{k2026group,
title = {{Group Petri Nets: Structural and Algebraic Models for Finite Groups and Their Subgroups}},
author = {Girish K P},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05846978v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Group Petri Nets: Structural and Algebraic Models for Finite Groups and Their Subgroups
AU - Girish K P
PY - 2026
DA - 2026/11/01
PB - HAL
UR - https://hal.science/hal-05846978v1
AB - Every finite group (G, * ) determines a Petri net N G whose places are the elements of G and whose transitions realise the group's Cayley table: transition t a,b consumes one token from a and one from b and deposits one token in a * b. We develop this correspondence as a single connected story. Restricting N G to a subgroup H ≤ G gives a Subgroup Petri Net; separating a group element's carrier and control roles gives a control-copy net M G ; and restricting the control layer of M G to H makes one-step reachability on the carrier layer coincide exactly with the rightcoset partition of G by H, giving a Petri-net reading of Lagrange's theorem, |G| = [G : H] • |H|. We show every Group Petri Net carries a strictly decreasing token-sum functional, forcing boundedness and termination of every firing sequence, with the terminal marking independent of firing order for every initial marking exactly when G is abelian. All constructions are illustrated with fully worked, diagrammed examples: Z 3 for the basic construction, and S 3 , with the explicit subgroup H = {e, (1 2)} and its three coset blocks, for the subgroup, control-copy, and non-confluence phenomena.
ER -