Is it true that: If two finite abelian groups of the same order have equal sum of element orders, then they are isomorphic?
I have tried getting a counterexample, but failed.
So as not to be misunderstood by the term "sum of element orders," I will explain with an example.
The sum of element orders of $\mathbb{Z}_2^2$ is $1+2+2+2=7$ while that of $\mathbb{Z}_4$ is $1+2+4+4=11$. This example shows that the sum of element orders of $\mathbb{Z}_2^2$ and $\mathbb{Z}_4$ are not equal.
Looking forward to suggestions on how to prove the above highlighted proposition.
Thanks.