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Groups with the same orders and large character degrees as \(\text{PGL}(2,9)\). (English) Zbl 1294.20009
Summary: An interesting class of problems in character theory arises from considering how the structure of a group \(G\) and the set \(\{\chi(1)\mid\chi\in\text{Irr}(G)\}\) are related. It is proved that some finite groups are uniquely determined by their character tables. Recently Xu, Chen and Yan proved that if \(G\) is a simple \(K_3\)-group, then \(G\) is uniquely determined by its order and one or both of its largest irreducible character degrees.
In this paper, we determine groups with the same order and the same largest and second largest irreducible character degrees as \(\text{PGL}(2,9)\) and as a consequence of our result it follows that \(\text{PGL}(2,9)\) is characterizable by the structure of its complex group algebra.

20C15 Ordinary representations and characters
20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)