论文标题
Gaschütz，Žmud$'$和Rhodes的定理扩展
Extensions of theorems of Gaschütz, Žmud$'$ and Rhodes on faithful representations
论文作者
论文摘要
Gaschütz（1954）证明了有限的$ g $具有忠实的不可约合的复杂表示，并且仅当其SOCLE是由单个元素作为正常子组产生的；只要$ g $没有非平凡的普通$ p $ -subgroup，该结果就扩展到了特征$ p $的任意字段。 Žmud$'$（1956）表明，在$ g $的忠实复杂代表中，最少的不可约成分数量与其SOCLE的最少发电机数量作为正常子组相吻合；该结果也可以扩展到任何特征$ p $的任意字段，以便$ g $没有非平凡的普通$ p $ -subgroup（即，$ g $在其上承认忠实的完全还原的表示）。 罗德（Rhodes）（1969）的特征是有限的半群岛，承认对任意领域的忠实不可减至的代表，因为广义的群体将半群落绘制在一个群体上，该小组承认对所涉及的领域的忠实不可减至的代表。在这里，我们通过确定在任意领域的有限半群的忠实降低表示中（前提是它具有一个），我们通过确定忠诚的有限半群的忠实代表中的最低数量的不可还原成分来提供对Žmud$'$和Rhodes定理的共同概括。 我们的Semigroup结果的关键工具是Žmud$'$'的定理的相对版本，该版本确定了有限的组$ g $和一个普通的亚组$ n \ lhd g $，在完全还原$ g $的限制对$ n $的限制中，最低数量是不可降低的成分数量。
Gaschütz (1954) proved that a finite group $G$ has a faithful irreducible complex representation if and only if its socle is generated by a single element as a normal subgroup; this result extends to arbitrary fields of characteristic $p$ so long as $G$ has no nontrivial normal $p$-subgroup. Žmud$'$ (1956) showed that the minimum number of irreducible constituents in a faithful complex representation of $G$ coincides with the minimum number of generators of its socle as a normal subgroup; this result can also be extended to arbitrary fields of any characteristic $p$ such that $G$ has no nontrivial normal $p$-subgroup (i.e., over which $G$ admits a faithful completely reducible representation). Rhodes (1969) characterized the finite semigroups admitting a faithful irreducible representation over an arbitrary field as generalized group mapping semigroups over a group admitting a faithful irreducible representation over the field in question. Here, we provide a common generalization of the theorems of Žmud$'$ and Rhodes by determining the minimum number of irreducible constituents in a faithful completely reducible representation of a finite semigroup over an arbitrary field (provided that it has one). Our key tool for the semigroup result is a relativized version of Žmud$'$'s theorem that determines, given a finite group $G$ and a normal subgroup $N\lhd G$, what is the minimum number of irreducible constituents in a completely reducible representation of $G$ whose restriction to $N$ is faithful.
