Recognizability of alternating groups by spectrum. (English. Russian original) Zbl 1272.20007
Algebra Logic 52, No. 1, 41-45 (2013); translation from Algebra Logika 52, No. 1, 57-63 (2013).
Summary: The spectrum of a group is the set of its element orders. A finite group \(G\) is said to be recognizable by spectrum if every finite group that has the same spectrum as \(G\) is isomorphic to \(G\). It is proved that simple alternating groups \(A_n\) are recognizable by spectrum, for \(n\neq 6,10\). This implies that every finite group whose spectrum coincides with that of a finite non-Abelian simple group has at most one non-Abelian composition factor.
MSC:
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D05 | Finite simple groups and their classification |
Keywords:
finite simple groups; alternating groups; spectra of groups; recognizability by spectrum; prime graphs; Gruenberg-Kegel graphs; sets of element orders; recognizable groupsReferences:
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