遗传根群类和半单群类

Hereditary Radical Classes and Semisimple Classes of Groups

针对Li X.M分别给出的根群类和半单群类的若干特征性质,提出了根群类和半单群类之间应具有的联系。通过群的一类特殊的正规子群:本质正规子群将根群类和半单群类联系起来,并利用群的根性的一般理论,得到了两个结果:根群类P是遗传的当且仅当其对应的半单群类L是本质扩张闭的;当根群类P是遗传根群类时,根群类P和其对应的半单群类L所决定的群类M是根群类。

Based on several characterizations of radical classes and semisimple classes of groups given respectively by Li X M, the relationship of radical classes and semisimple classes of groups is proposed. Linked to radical classes of groups and semisimple classes of groups by a special normal subgroup of a group, the essential normal subgroup, and by using the general theory of radical properties of groups, we got the following two results. A radical class P of groups is hereditary if and only if the corresponding semisimple class L of groups is closed under essential extension. If a radical class P is a hereditary radical class of groups, then the class M of groups determined by P and the corre- sponding semisimple class L is a radical class of groups.