Subgroup-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 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 
that sends each group 
to a subgroup 
. By subgroup here we mean an abstract group along with an embedding into . The subgroup-defining function should satisfy the property that whenever there is an isomorphism 
→ 
, the image of under 
is 
.
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 
, for a fixed prime , that returns nontrivial subgroups for nontrivial groups, is termed a characteristic p-functor. Characteristic p-functors are a special case of conjugacy functors.
Property theory
Subgroup properties satisfied
Further information: subgroup-defining function value is characteristic
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 , it also always satisfies the property intermediately . In particular, any subgroup obtained this way is an intermediately characteristic subgroup.
Associated constructions
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 , gives 
.
We can continue this series transfinitely by defining the 
term as of the 
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 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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.
Operators to group properties
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.
