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Suppose G {\displaystyle G} is a finite group and c {\displaystyle c} is a positive integer. Define the set:
C T c ( G ) = { ( x 1 , x 2 , … , x c + 1 ) ∈ G c + 1 ∣ [ [ … [ x 1 , x 2 ] , x 3 ] , … , x c ] , x c + 1 ] = e } {\displaystyle 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 {\displaystyle c} tuple fraction of G {\displaystyle G} is defined as:
| C T c ( G ) | | G | c + 1 {\displaystyle {\frac {|CT_{c}(G)|}{|G|^{c+1}}}}
Note that this fraction equals 1 if and only if G {\displaystyle G} is a nilpotent group of nilpotency class at most c {\displaystyle c} .
is a finite group and 
is a positive integer. Define the set:
![{\displaystyle 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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11a7cd52d9cb48657641e3b1d8945391c5d50659)
tuple fraction of is defined as:
is a nilpotent group of nilpotency class at most .