论文标题
有完美订单课程的汉密尔顿团体
Hamiltonian Groups with Perfect Order Classes
论文作者
论文摘要
如果任何给定订单的要素的数量为零，则有限组具有“完美的订单类”。 本说明的目的是用完美的订单课程明确描述有限的哈密顿群体。我们表明，有限的哈密顿群集团有完美的订单课程，并且只有当它与四订单订单$ 8 $的直接产品同构时，一个非平凡的循环$ 3 $ - 组和最多2美元的订单。 定理。有限的汉密尔顿组有完美的订单类，并且只有在$ q \ times c_ {3^k} $或$ q \ q \ times c_ {2} \ times c_ {3^k} $中，对于某些正整数$ k $而言。
A finite group is said to have "perfect order classes" if the number of elements of any given order is either zero or a divisor of the order of the group. The purpose of this note is to describe explicitly the finite Hamiltonian groups with perfect order classes. We show that a finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic to the direct product of the quaternion group of order $8$, a non-trivial cyclic $3$-group and a group of order at most $2$. Theorem. A finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic either to $Q\times C_{3^k}$ or to $Q\times C_{2}\times C_{3^k}$, for some positive integer $k$.
