Let be a finite abelian group of type . (That is, are the invariant factors of .) Prove that contains an element of order if and only if . Deduce that is of exponent .
Suppose contains an element of order . Now for some , and we have . By the divisibility criterion for invariant factors, is a common multiple of the , hence of the . Thus .
Suppose . Now has a subgroup isomorphic to , which (being cyclic) has an element of order . Thus has an element of order .
We conclude that if has order , then since for some , . Hence the exponent of is finite and divides . Moreover, the exponent of is at least since has an element of order . Thus has exponent .