Prime Graphs of Locally Torsion Ultragroups, Metanilpotent Groups, and T-Solvable Groups

Abstract

Given a group G, the prime graph of G, denoted Γ(G), is the graph with vertex set the set of primes p such that there exists an element of order p in G, and such that there is an edge between primes p and q if and only if there exists an element of order pq in G. Let T be nonabelian finite simple group. A T-solvable group G is a group such that there exists a finite subnormal series of G such that each factor is either abelian or isomorphic to T. In this paper, we continue the classification of the prime graphs of T-solvable locally torsion groups and classify prime graphs of locally torsion metanilpotent groups using tools from model theory. We also classify the prime graphs of ultragroups with certain first-order properties and prove results on the number and type of certain classes of groups that can realize a given graph as a prime graph.

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BibTeX RIS

BibTeX

@misc{renner2026prime,
  title = {{Prime Graphs of Locally Torsion Ultragroups, Metanilpotent Groups, and $T$-Solvable Groups}},
  author = {Alexa Renner},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02696},
  eprint = {2610.02696},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02696},
}

RIS

TY  - UNPB
TI  - Prime Graphs of Locally Torsion Ultragroups, Metanilpotent Groups, and T-Solvable Groups
AU  - Alexa Renner
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02696
UR  - https://arxiv.org/abs/2610.02696
AB  - Given a group G, the prime graph of G, denoted Γ(G), is the graph with vertex set the set of primes p such that there exists an element of order p in G, and such that there is an edge between primes p and q if and only if there exists an element of order pq in G. Let T be nonabelian finite simple group. A T-solvable group G is a group such that there exists a finite subnormal series of G such that each factor is either abelian or isomorphic to T. In this paper, we continue the classification of the prime graphs of T-solvable locally torsion groups and classify prime graphs of locally torsion metanilpotent groups using tools from model theory. We also classify the prime graphs of ultragroups with certain first-order properties and prove results on the number and type of certain classes of groups that can realize a given graph as a prime graph.
ER  -

Abstract

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