# If two finite groups have the same number of elements of the same order, does this necessarily imply they are isomorphic? [duplicate]

I am trying to prove a rather difficult question and I have arrived at a small proof that I can prove true after thoroughly (and exhaustively) analyzing the group structure.

My question is, if two finite groups have the same number of elements of the same order... are they necessarily isomorphic? If not, what are some properties of the group structure that can show that two groups are necessarily isomorphic?

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In general, finite groups of the same order are not isomorphic. Indeed, this is far from true. So there are, for example, 49 487 365 422 pairwise non-isomorphic groups of order $1024$. This is a lot.