论文标题
计算一些F组的中心化和Z级
Counting centralizers and z-classes of some F-groups
论文作者
论文摘要
有限的组$ g $称为f组，如果每$ x，y \ in g \ setminus z（g）$，$ c（x）\ leq c（y）$表示$ c（x）= c（y）$。另一方面，据说一个组的两个要素是$ z $等效的，或者在同一$ z $ class中，如果他们的中央化合物在该组中是偶然的。在本文中，对于一个有限的组，我们为中央位置数量/ $ z $类别提供了必要和足够的条件，以等于其中心的索引。我们还为有限的F组的$ z $类数量提供了必要的条件，以达到其最大数量（这扩展了更早的结果）。除其他结果外，我们计算了某些有限组的元素中心化和$ z $类别的数量，并扩展了一些先前的结果。
A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their centralizers are conjugate in the group. In this paper, for a finite group, we give necessary and sufficient conditions for the number of centralizers/ $z$-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of $z$-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and $z$-classes of some finite groups and extend some previous results.
