标题： 有限群代数的阿贝尔-循环群的正则补问题 显示英文标题

标题： On the normal complement problem for finite group algebras of Abelian-by-cyclic groups

摘要： 假设$F$是一个阶为$p^f$的有限域，$q$是一个奇素数，其中$p^f-1=sq^m$，其中$m \ge 1$和$(s,q)=1$。 在本文中，我们得到了半单群代数$FC_q.$的对称和酉子群的阶。 进一步地，对于由阿贝尔群$A$通过$C_q = \langle b \rangle$扩展得到的$G$，其阶为$p^n$且满足$C_{A}(b) = \{e\}$，我们证明了如果$m>1,$或$(s+1) \geq q$且$2n \geq f(q-1)$，则$G$在$V(FG)$中没有正规补。 显示英文摘要

摘要： Assume $F$ is a finite field of order $p^f$ and $q$ is an odd prime for which $p^f-1=sq^m$, where $m \ge 1$ and $(s,q)=1$. In this article, we obtain the order of symmetric and unitary subgroup of the semisimple group algebra $FC_q.$ Further, for the extension $G$ of $C_q = \langle b \rangle$ by an abelian group $A$ of order $p^n$ with $C_{A}(b) = \{e\}$, we prove that if $m>1,$ or $(s+1) \geq q$ and $2n \geq f(q-1)$, then $G$ does not have a normal complement in $V(FG)$.