The number of homomorphisms from the free abelian group of rank r into a finite group G grows asymptotically as k * m^r, where m is the order of the largest abelian subgroup and k is the number of such subgroups.
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Higher Commutativity in Finite Groups: Exact Asymptotics and Finite Spectrum
The number of homomorphisms from the free abelian group of rank r into a finite group G grows asymptotically as k * m^r, where m is the order of the largest abelian subgroup and k is the number of such subgroups.