Recognition of the symplectic simple group $ PSp_4(p) $ by the order and degree prime-power graph

Abstract

Let $ G $ be a finite group, $ \operatorname{cd}(G) $ the set of all irreducible character degrees of $ G $, and $ \rho(G) $ the set of all prime divisors of integers in $ \operatorname{cd}(G) $. For a prime $ p $ and a positive integer $ n $, let $ n_p $ denote the $ p $-part of $ n $. The degree prime-power graph of $ G $ is a graph whose vertex set is $ V(G) = \left\{p^{e_p(G)} \mid p \in \rho(G)\right\} $, where $ p^{e_p(G)} = \max \left\{n_p \mid n \in \operatorname{cd}(G)\right\} $, and there is an edge between distinct numbers $ x, y \in V(G) $ if $ x y $ divides some integer in $ \operatorname{cd}(G) $. The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group $ PSp_4(p) $ can be uniquely identified by its order and degree prime-power graph.

Keywords

• degree prime-power graph
• irreducible character
• degree
• maximal $ p $-part
• simple group
• subnormal series