关于群中两可交换元素乘积的阶

A Note on the Order of a Product of Communitive Elements in a Group

在交换群中,给出两元素乘积的阶不等于它们阶的最小公倍数的例子;讨论两元素乘积的阶等于它们阶的最小公倍数的充要条件;证明了若两个交换p-元素乘积的阶小于它们阶的最小公倍数,那么这两个p-元素的阶相等.

In the present note, suppose in a group that two elements of finite order are commutative with each other, we give examples to show that the order of their product does not eaqual to the 1. c. m of their orders. And we get an if-and-only-if condition for two finite elements, with the order of their product equals to the 1. c. m of their orders. We prove that, for communicated p-elements α, b, if the order of αb does not eaqual to the l. c. m of the o （ α ） and o （ b ）, then the order of these two elements o （ α ） and o （ b ） must be the same.