Prime Graphs of Infinite Groups
Abstract
The prime graph of a finite group G is the graph Γ(G) with vertex set the set of prime divisors π(G) of |G| and an edge between vertices p, q∈π(G) if and only if there exists an element g∈ G with order o(g) = pq. Given a finite nonabelian simple group T, a group G is T-solvable if there exists a composition series of G such that every composition factor is either abelian or isomorphic to T. In this paper, we introduce the prime graph of an infinite group, the graph Γ(G) with vertex set π(G) = {o(g):g∈ G and o(g) is prime} and an edge between p,q∈π(G) if and only if there exists an element g∈ G with order pq, and generalize several results on the prime graphs of finite solvable and T-solvable groups to results on the prime graphs of members of certain classes of infinite solvable and T-solvable groups.
Related
- Prime-valent Symmetric Cayley Graphs of Characteristically Simple Groups
- Prime Graphs of Locally Torsion Ultragroups, Metanilpotent Groups, and T-Solvable Groups
- Alternating Divisoriality of Prime Ideals along Infinite Chains of Bézout Domains
- Infinite prime sumsets in structured and Uk(Φ)-uniform sets
- Fast computation and convergence analysis of the infinite-product representation of the Schottky--Klein prime function
BibTeX
@misc{renner2026prime,
title = {{Prime Graphs of Infinite Groups}},
author = {Alexa Renner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02102},
eprint = {2610.02102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02102},
}
RIS
TY - UNPB
TI - Prime Graphs of Infinite Groups
AU - Alexa Renner
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02102
UR - https://arxiv.org/abs/2610.02102
AB - The prime graph of a finite group G is the graph Γ(G) with vertex set the set of prime divisors π(G) of |G| and an edge between vertices p, q∈π(G) if and only if there exists an element g∈ G with order o(g) = pq. Given a finite nonabelian simple group T, a group G is T-solvable if there exists a composition series of G such that every composition factor is either abelian or isomorphic to T. In this paper, we introduce the prime graph of an infinite group, the graph Γ(G) with vertex set π(G) = {o(g):g∈ G and o(g) is prime} and an edge between p,q∈π(G) if and only if there exists an element g∈ G with order pq, and generalize several results on the prime graphs of finite solvable and T-solvable groups to results on the prime graphs of members of certain classes of infinite solvable and T-solvable groups.
ER -