On the probability distribution associated to commutator word map in finite groups \rom{2}

Abstract: Let

P (G)

denotes the set of sizes of fibers of non-trivial commutators of the commutator word map. Here, we prove that

| P (G) | = 1

, for any finite group

G

of nilpotency class

3

with exactlly two conjugacy class sizes. We also show that for given

n g e q 1

, there exists a finite group

G

of nilpotency class

2

with exactlly two conjugacy class sizes such that

| P (G) | = n

.

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