论文标题
在两个有限的不可溶解组的Möbius功能上
On two Möbius function for a finite non-solvable group
论文作者
论文摘要
令$ g $为有限的组，$μ$是$ g $的亚组晶格上的莫比乌斯功能，而$λ$是$ g $的子组的偶数类别的莫比乌斯函数。 Pahlings证明，每当$ g $可解决时，属性$μ（h，g）= [n_ {n_ {g^\ prime}（h）：g^{\ prime} \ cap h] \cdotλ（h，g）$持有任何子组$ h $ g $ $ g $。众所周知，这个属性一般不存在。例如，它并不适用于每个简单的组，Mathieu组$ m_ {12} $是反例。在本文中，我们研究了某些不可解决的组的$ $ $ $和$λ$之间的关系；其中，最小的不可溶解群体。我们还提供了几个不满足该财产的群体的例子。
Let $G$ be a finite group, $μ$ be the Möbius function on the subgroup lattice of $G$, and $λ$ be the Möbius function on the poset of conjugacy classes of subgroups of $G$. It was proved by Pahlings that, whenever $G$ is solvable, the property $μ(H,G)=[N_{G^\prime}(H):G^{\prime}\cap H]\cdotλ(H,G)$ holds for any subgroup $H$ of $G$. It is known that this property does not hold in general; for instance it does not hold for every simple groups, the Mathieu group $M_{12}$ being a counterexample. In this paper we investigate the relation between $μ$ and $λ$ for some classes of non-solvable groups; among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.
