I just finished below to exercises.
If $G$ is Abelian group of order $9$ then, maximum number of elements (excluding the identity)that one need to compute the order, to determine the isomorphism class of $G$ is $3$
If $G$ is Abelian group of order $18$ then, maximum number of elements (excluding the identity)that one need to compute the order, to determine the isomorphism class of $G$ is $6$
My Question: Is there is general statement regarding this?
I mean,
If $G$ is Abelian group of order $n$ then, what is the maximum number of elements (excluding the identity)that one need to compute the order, to determine the isomorphism class of $G$?
Please help