Realizing Gruenberg-Kegel graphs of $T$-solvable groups with structurally simplified extensions of $T$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914165618114560 |
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| author | Alland, Lucas Fridman, Andrei Keller, Thomas Michael |
| author_facet | Alland, Lucas Fridman, Andrei Keller, Thomas Michael |
| contents | Given a finite group $G$, its prime graph $Γ(G)$ (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of $|G|$ and where edges $\{p, q\}$ exist whenever $G$ contains an element of order $pq$. We continue the study of prime graphs for $T$-solvable groups; that is, groups whose composition factors are either abelian or isomorphic to some fixed non-abelian simple group $T$. For a large class of non-abelian simple groups $T$, we prove that the prime graph complements of $T$-solvable groups are always realizable by a solvable group and a quasi simple or almost simple $T$-solvable group acting by automorphisms on a direct product of elementary abelian groups. We conjecture that a similar result holds in full generality. Moreover, we apply our result to classify in purely graph-theoretic terms the prime graph complements of $\operatorname{PSL}(2,13)$-solvable groups, and indicate other interesting classes of groups matching the assumptions of our main theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Realizing Gruenberg-Kegel graphs of $T$-solvable groups with structurally simplified extensions of $T$ Alland, Lucas Fridman, Andrei Keller, Thomas Michael Group Theory 20D60, 05C25 Given a finite group $G$, its prime graph $Γ(G)$ (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of $|G|$ and where edges $\{p, q\}$ exist whenever $G$ contains an element of order $pq$. We continue the study of prime graphs for $T$-solvable groups; that is, groups whose composition factors are either abelian or isomorphic to some fixed non-abelian simple group $T$. For a large class of non-abelian simple groups $T$, we prove that the prime graph complements of $T$-solvable groups are always realizable by a solvable group and a quasi simple or almost simple $T$-solvable group acting by automorphisms on a direct product of elementary abelian groups. We conjecture that a similar result holds in full generality. Moreover, we apply our result to classify in purely graph-theoretic terms the prime graph complements of $\operatorname{PSL}(2,13)$-solvable groups, and indicate other interesting classes of groups matching the assumptions of our main theorem. |
| title | Realizing Gruenberg-Kegel graphs of $T$-solvable groups with structurally simplified extensions of $T$ |
| topic | Group Theory 20D60, 05C25 |
| url | https://arxiv.org/abs/2511.16403 |