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A characterization of some alternating groups by their orders and character degree graphs. (English) Zbl 1368.20006
Summary: The aim of this study was to characterize some alternating groups by their orders and character degree graphs. To achieve this, $$G$$ was used as a finite group. The character degree graph $$\Gamma(G)$$ of $$G$$ is the graph whose vertices are the prime divisors of character degrees of $$G$$, and two vertices $$p$$ and $$q$$ are joined by an edge if $$p\cdot q$$ divides some character degree of $$G$$. $$A_n$$ was used as an alternating group of degree $$n$$. B. Khosravi et al. [Miskolc Math. Notes 15, No. 2, 537–544 (2014; Zbl 1324.20004)] have shown that $$A_n$$, with $$n=5,6,7$$ are characterizable by the character degree graphs and their orders. The results of this study achieved the conclusion of characterizing the alternating group $$A_n$$, where $$n=8,9,10$$, by using its character degree graph and order. In particular, the alternating groups $$A_9$$ and $$A_{10}$$ are not unique determined by their character degree graphs and their orders.
##### MSC:
 20C15 Ordinary representations and characters 20C33 Representations of finite groups of Lie type 20D06 Simple groups: alternating groups and groups of Lie type 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups