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 $f$ 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 $\sigma :G$ → $H$, the image of $f(G)$ under $\sigma$ is $f(H)$.

Comparison with concept of functor in category theory

A subgroup-defining function is not a functor on the category of groups as a whole, because it is not guaranteed to play well with all group homomorphisms. However, it does play well with isomorphisms, so it is a functor in that category. However, if we were to just consider the subcategory of the category of groups with isomorphisms, then we lose the information that encode subgroup inclusion, so the "subgroup-defining" part becomes unenforceable.

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.

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, it defines a verbal subgroup and hence a fully invariant 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.

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 $G$, 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 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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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.