Nilpotency class and order determine group up to commutator map-equivalence for up to prime-fourth order

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This article gives a fact that is true for small groups of prime power order.More specifically, it is true for all groups of order p^k where k is at most equal to 4.
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Suppose k and c are natural numbers and 0 \le k \le 4. Suppose p is a prime number. Then, if G,H are groups both of order p^k and nilpotency class exactly c, G and H must be Commutator map-equivalent groups (?).

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