Quotient-defining function

Symbol-free definition
A quotient-defining function is a rule that sends each group to a quotient group, and such that any isomorphism of groups take the defined quotient of one group to the defined quotient for the other.

Definition with symbols
A quotient-defining function is a rule $$f$$ that sends each group $$G$$ to a quotient $$f(G)$$. By subgroup here we mean an abstract group along with a surjective homomorphism from $$G$$. The quotient-defining function should satisfy the property that whenever there is an isomorphism $$\sigma: G \to H$$, the image of $$f(G)$$ under $$\sigma$$ is $$f(H)$$.

The kernel is a subgroup-defining function
Given any quotient-defining function, we can consider an associated subgroup-defining function, viz the subgroup-defining function that sends a given group to the kernel for that quotient group.

Conversely, any subgroup-defining function gives a corresponding quotient-defining function, viz the function that sends a given group to its quotient by that subgroup. Note that the quotient by that subgroup is well-defined because any subgroup-defining function returns a characteristic subgroup and every characteristic subgroup is normal.

The image operator
The image operator is a map from the collection of quotient-defining functions to the collection of group properties that sends a quotient-defining function to the property of being a group that can arise as the result of applying the quotient-defining function to some group.

The fixed-point operator
The fixed-point operator is a map from the collection of quotient-defining functions to the collection of group properties that sends a quotient-defining function to the property of being a group which is fixed point under this subgroup-defining function.

For idempotent quotient-defining functions, the fixed-point operator and image operator have the same effect.

The free operator
The free operator is a map from the collection of quotient-defining functions to the collection of group properties. This sends a quotient-defining function to the property of being a group for which the corresponding subgroup is the trivial quotient.