Quotient-defining function

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This article is about a general term. A list of important particular cases (instances) is available at Category: Subgroup-defining functions

Definition

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 $f(H)$.

Property theory

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.

Operators to group properties

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.