标题： 完美数和群 显示英文标题

标题： Perfect numbers and groups

摘要： 一个数是完美的，如果它是其真因数的和；在这里，我们称一个有限群为“完美”，如果它的阶是其真正规子群的阶的和。 （这与标准术语冲突，但不会引起混淆。） “完美群”的概念推广了“完美数”的概念，因为当且仅当其阶是完美数时，循环群才是完美的。 我们证明实际上，唯一的阿贝尔完美群是循环群，并展示了一些偶阶的非阿贝尔完美群。 显示英文摘要

摘要： A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not arise.) The notion of perfect group generalizes that of perfect number, since a cyclic group is perfect exactly when its order is perfect. We show that, in fact, the only abelian perfect groups are the cyclic ones, and exhibit some non-abelian perfect groups of even order.