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A new characterization of \(A_{12}\). (English) Zbl 1270.20012

Summary: Purpose: It is well known that the conjugacy class sizes have an important influence on the structure of a group. This work is considering a different set of ‘sizes’, the number of elements of a given order.
Methods: By using the set \(\text{nse}(G)\) and the order of \(G\), we prove that \(G\) is isomorphic to \(A_{12}\).
Results: Thompson’s conjecture is true for \(A_{12}\).
Conclusions: We prove that a finite group \(G\) is isomorphic to \(A_{12}\), the alternating group \(A_{12}\) of degree 12 if, and only if, \(|G|=|A_{12}|\) and \(\text{nse}(G)=\text{nse}(A_{12})\).

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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