A characterization of some alternating groups by their orders and character degree graphs.

*(English)*Zbl 1368.20006Summary: 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 |