Subgroup-defining function

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

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

For particular kinds of groups
Subgroup-defining functions of particular kinds on groups of prime power order are particularly important. A subgroup-defining function on all groups whose order is a power of $$p$$, for a fixed prime $$p$$, that returns nontrivial subgroups for nontrivial groups, is termed a characteristic p-functor. Characteristic p-functors are a special case of conjugacy functors.

Subgroup properties satisfied
Any subgroup-defining function defines a characteristic subgroup. If the function is a verbal function, its defines a verbal subgroup and hence a fully characteristic subgroup, whereas if the function is a word-bound function, it defines a word-bound subgroup and hence a strictly characteristic subgroup.

If the subgroup-defining function is monotone and idempotent, then it is intermediacy-preserved. Thus, if it always satisfies a subgroup property $$p$$, it also always satisfies the property intermediately $$p$$. In particular, any subgroup obtained this way is an intermediately characteristic subgroup.

Quotient-defining function
To any subgroup-defining function, we can also associate a corresponding quotient-defining function, which sends a group to the quotient group by that subgroup. Note that the quotient group is well-defined because a subgroup-defining function always returns a characteristic subgroup, and every characteristic subgroup is normal.

Descending series
Given a subgroup-defining function, the associated descending series, also called the series obtained by iteration of the subgroup-defining function, is the series that, for any group $$G$$, gives $$G, f(G), f(f(G))...$$.

We can continue this series transfinitely by defining the $$(\alpha +1)^{th}$$ term as $$f$$ of the $$\alpha^{th}$$ term, and by defining the term for a limit ordinal as the intersection of all its predecessors.

As with any descending series-defining function, we can talk of the length, and the notion of being descendable to the trivial subgroup. Thus, we can associate these notions to any subgroup-defining function.

Ascending series
Given a subgroup-defining function, the associated ascending series, also called the series obtained by quotient-iteration, is defined as

As with any ascending series-defining function, we can talk of the length, and the notion of being ascendable to the whole group. Thus, we can associate these notions to any subgroup-defining function.

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

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

For idempotent subgroup-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 subgroup-defining functions to the collection of group properties. This sends a subgroup-defining function to the property of being a group for which the corresponding subgroup is the trivial subgroup.