论文标题
在每个元素都有主要功率顺序并满足某些有限条件的组上
On groups in which every element has a prime power order and which satisfy some boundedness condition
论文作者
论文摘要
在本文中，我们将与定期组打交道，其中每个元素都具有主要的电源顺序。如果$ g $的每个元素都有质量电源订单，并且对于π（g）$中的每个$ p \，则一个$ bcp $ - 组将被称为$ bcp $ - 组，则存在一个正整数$ u_p $，因此每个$ p $ element $ g $均为$ g $ is of $ p $ p^i \ p^i \ leq p^{u_p} $。如果$ g $的每个元素都有主要的电源订单，并且对于π（g）$中的每个$ p \，则组$ g $将被称为$ bsp $ - 组。在这里，$π（g）$表示所有素数的集合，将$ g $的某些元素的顺序分开。我们的主要结果是以下四个定理。定理1：让$ g $是有限生成的$ bcp $ -group。然后，$ G $只有有限的有限指数子组有限的子组。定理4：让$ g $是本地分级的$ bcp $ -group。然后$ g $是本地有限的组。定理7：让$ g $是本地分级$ bsp $ group。然后$ g $是有限的集团。定理9：让$ g $为$ bsp $ - 群体满足$ 2 \ inπ（g）$。然后$ g $是本地有限的组。
In this paper we shall deal with periodic groups, in which each element has a prime power order. A group $G$ will be called a $BCP$-group if each element of $G$ has a prime power order and for each $p\in π(G)$ there exists a positive integer $u_p$ such that each $p$-element of $G$ is of order $p^i\leq p^{u_p}$. A group $G$ will be called a $BSP$-group if each element of $G$ has a prime power order and for each $p\in π(G)$ there exists a positive integer $v_p$ such that each finite $p$-subgroup of $G$ is of order $p^j\leq p^{v_p}$. Here $π(G)$ denotes the set of all primes dividing the order of some element of $G$. Our main results are the following four theorems. Theorem 1: Let $G$ be a finitely generated $BCP$-group. Then $G$ has only a finite number of normal subgroups of finite index. Theorem 4: Let $G$ be a locally graded $BCP$-group. Then $G$ is a locally finite group. Theorem 7: Let $G$ be a locally graded $BSP$-group. Then $G$ is a finite group. Theorem 9: Let $G$ be a $BSP$-group satisfying $2\in π(G)$. Then $G$ is a locally finite group.
