Fixed-class tuple fraction

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Definition

Suppose G is a finite group and c is a positive integer. Define the set:

CT_c(G) = \{ (x_1,x_2,\dots,x_{c+1}) \in G^{c+1} \mid [[ \dots [x_1,x_2],x_3],\dots,x_c],x_{c+1}] = e \}

Then, the class c tuple fraction of G is defined as:

\frac{|CT_c(G)|}{|G|^{c+1}}

Note that this fraction equals 1 if and only if G is a nilpotent group of nilpotency class at most c.

Facts