论文标题
简单组的亚组尽可能多样化
Subgroups of simple groups are as diverse as possible
论文作者
论文摘要
对于有限的组$ g $，令$σ（g）$是$ g $和$σ_ι（g）$的亚组的子组的数量，$ g $的亚组的同构类型类型。 令$ l = l_r（p^e）$表示一个简单的谎言类型，排名$ r $，在订单$ p^e $和特征$ p $的字段上。如果$ r \ neq 1 $，$ l \ not \ chan $$ p^{（c-o（1））r^4e^2} \leqσ_ι（l_r（p^e））\leqσ（l_r（p^e））\ leq p^{（d+o（d+o（1））r^4e^2}。对于其他古典组$ 1/64 \ leq c \ leq d \ leq 1/4 $。对于特殊和扭曲的组$ 1/2^{100} \ leq c \ leq d \ leq 1/4 $。此外，$$ 2^{（1/36-o（1））k^2）} \leqσ_ι（\ m atrm {alt} _k）\ leqσ（\ mathrm {alt} _k）\ leq 24^{（1/6+o（1/6+o（1/6+o（1/6+o（1/6+o）， $σ_ι（g），σ（g）\ in O（1）$。通常，在相同订单的组中，最好的这些范围是最好的。因此，除了具有有限等级和场度的有限简单组外，有限简单组的亚组尽可能多样化。
For a finite group $G$, let $σ(G)$ be the number of subgroups of $G$ and $σ_ι(G)$ the number of isomorphism types of subgroups of $G$. Let $L=L_r(p^e)$ denote a simple group of Lie type, rank $r$, over a field of order $p^e$ and characteristic $p$. If $r\neq 1$, $L\not\cong {^2 B_2}(2^{1+2m})$, then there are constants $c,d$, dependent on the Lie type, such that as $re$ grows $$p^{(c-o(1))r^4e^2}\leqσ_ι(L_r(p^e))\leqσ(L_r(p^e)) \leq p^{(d+o(1))r^4e^2}.$$ For type $A$, $c=d=1/64$. For other classical groups $1/64\leq c\leq d\leq 1/4$. For exceptional and twisted groups $1/2^{100}\leq c\leq d\leq 1/4$. Furthermore, $$2^{(1/36-o(1))k^2)}\leqσ_ι(\mathrm{Alt}_k)\leq σ(\mathrm{Alt}_k)\leq 24^{(1/6+o(1))k^2}.$$ For abelian and sporadic simple groups $G$, $σ_ι(G),σ(G)\in O(1)$. In general these bounds are best possible amongst groups of the same orders. Thus with the exception of finite simple groups with bounded ranks and field degrees, the subgroups of finite simple groups are as diverse as possible.
