Characterization of $L_2(16)$ by $\tau_e(L_2(16))$

DOI：10.3770/j.issn:2095-2651.2012.02.013

 作者 单位 张庆亮 南通大学理学院 江苏 南通 226007 施武杰 重庆文理学院数学与统计学院 重庆 永川 402160

设$G$是群, $\phi_e(G)$为$G$中元的阶之集.又设$k\in \pi_e(G)$, $m_k$为$G$中$k$阶元的个数.令$\tau_e(G)=\{m_k |k\in \pi_e(G)\}$.本文中,我们证明了$L_2(16)$可用$\tau_e(L_2(16))$刻画.换句话说,我们证明了: 若$G$是群,使得 $\tau_e(G)=\tau_e(L_2(16))=\{1, 255, 272, 544, 1088, 1920\}$,则$G$同构于 $L_2(16)$.

Let $G$ be a group and $\pi_e(G)$ the set of element orders of $G$. Let $k\in \pi_e(G)$ and $m_k$ be the number of elements of order $k$ in $G$. Let $\tau_e(G)=\{m_k |k\in \pi_e(G)\}$. In this paper, we prove that $L_2(16)$ is recognizable by $\tau_e(L_2(16))$. In other words, we prove that if $G$ is a group such that $\tau_e(G)=\tau_e(L_2(16))=\{1, 255, 272, 544, 1088, 1920\}$, then $G$ is isomorphic to $L_2(16)$.