Classification about Non-Solvable Groups with Exactly 40 Maximal Order Elements

Abstract:

      Let $\varphi$ be a homomorphism from a group $H$ to a group ${\rm Aut}(N)$. Denote by $H_{\varphi}\times N$ the semidirect product of $N$ by $H$ with homomorphism $\varphi$. This paper proves that: Let $G$be a finite nonsolvable group. If $G$ has exactly 40 maximal order elements, then $G$ is isomorphic to one of the following groups: (1)~$Z_{4\varphi}\times A_5$,\,${\rm ker}\varphi=Z_2$; (2)~$D_{8\varphi}\times A_5,\,{\rm ker}\varphi=Z_2\times Z_2$; (3)~$G/N=S_5$, $N=Z(G)=Z_2$; (4)~$G/N=S_5$, $N=Z_2\times Z_2,\,N\cap Z(G)=Z_2$.