Subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category: Subgroup properties
Definition
Symbol-free definition
A subgroup property is a map from the collection of all group-subgroup pairs to the two-element set (true, false), with the property that two equivalent subgroups (viz equivalent group-subgroup pairs) either both have the property or both do not have the property. The subgroup property space is the collection of all subgroup properties.
Definition with symbols
A subgroup property is defined as a map that takes as input a pair
(where
is a subgroup of
) and outputs either True or False in such a way that if
≤
and
≤
are equivalent as group-subgroup pairs then:
satisfies
if and only if
does.
We say loosely that satisfies
in
or that
is a subgroup with property
in
.
Caution
Whether a subgroup satisfies the property in the given group cannot be completely decided by looking at the isomorphism class of the group and the subgroup -- the manner in which the subgroup sits inside the group is also important.
Examples
Important examples of subgroup properties
Roughly speaking, the idea behind a subgroup property is that given any group, and any subgroup, the subgroup either satisfies the property inside the group or it does not satisfy. A natural example of a subgroup property is the property of being a normal subgroup.
Given any subgroup of a group
, we either have that
is normal in
or that
is not normal in
.
Other important subgroup properties are such as the property of being a maximal subgroup, a maximal normal subgroup, a characteristic subgroup and so on.
Structure of the subgroup property space
Partial order
As for any property space, there is a natural partial order on the subgroup property space. Given subgroup properties and
, we say that
is stronger than
, or
is smaller than
, or
implies
, if whenever a subgroup satisfies property
in the whole group, it also satisfies property
in the whole group.
The tautology, fallacy, trivial and improper properties
The two extreme elements of the subgroup property space with respect to the partial order are the tautology -- the property satisfied by all subgroups, and the fallacy -- the property satisfied by no automorphism.
Apart from these, there are two other important properties:
- The [[improper property: This is the property satisfied by any group as a subgroup of itself.
- The trivial property: This is the property satisfied by the trivial subgroup in any group.
A subgroup property that implies the improper property is termed identity-true, and a subgroup property that implies the trivial property is termed trivially true. A subgroup property that is both trivially true and identity-true is termed trim.
Conjunction of subgroup properties
Given a collection of subgroup properties, the conjunction of those subgroup properties is defined as the property that a subgroup satisfies if and only if it satisfies all the given subgroup properties. Thus, conjunction is the analogue of logical conjunction, and of set-theoretic intersection. Note, however, that conjunction is very different from the intersection operator on subgroup properties.
Disjunction of subgroup properties
Given a collection of subgroup properties, the disjunction of those subgroup properties is defined as the property that a subgroup satisfies if and only if it satisfies at least one of the given subgroup properties. Thus, disjunction is the analogue of logical disjunction, or of set-theoretic union. Note, however, that disjunction is very different from the join operator on subgroup properties.
Operators on the subgroup property space
A full list is available at Category: Subgroup property operators.
The composition operator
Further information: composition operator
The composition operator on the subgroup property space is an associative binary operation from the subgroup property space to itself, which sends a pair of properties and
to the property
such that:
A subgroup satisfies property
in a group
if there is a subgroup
between
and
such that
satisfies
in
and
satisfies
as a subgroup of
.
The identity element for the composition operator is the identity property. The nil element for the composition operator is the fallacy. Since the composition operator is associative, it admits notions of transiters with the transiter master theorems holding.
The upper and lower hook operators
The upper and lower hook operators arise from the same idea of composition but with different inputs and different outputs.
Particular cases of these operators are:
- The potentially operator
- The intermediately operator
- The left hereditarily operator
- The right hereditarily operator
- The upward-closed operator
The intersection operator
Further information: intersection operator
The intersection operator sends a pair of properties and
to the property
such that:
A subgroup of a group
satisfies
in
if and only if there exist subgroups
and
of
such that
∩
,
satisfies
and
satisfies
.
The join operator
Further information: join operator
The join operator sends a pair of properties and
to the property
such that:
A subgroup of a group
satisfies
in
if and only if there exist subgroups
and
of
such that
satisfies
,
satisfies
, and
is the subgroup generated by
and
.
Formalisms for subgroup properties
The function restriction formalism
Further information: function restriction formalism
The function restriction formalism expresses subgroup properties that can be described in terms of the way they relate functions from the group to itself with functions from the subgroup to itself. More precisely, let and
be two function properties. Then:
→
is the subgroup property defined as follows: every function from the group to itself satisfying restricts to a function from the subgroup to itself satisfying
.
Such an expression is termed a function restriction formal expression. Function restriction formal expressions are available for properties such as being normal, characteristic, and being a central factor. These formal expressions allow various manipulations.
The relation implication formalism
Further information: relation implication operator
The relation implication formalism (or relation implication operator) takes as input two subgroup relations and
and outputs the property of being a subgroup such that any subgroup to which it is related via
is also such that they are related via
(if the relation is not transitive, we have to be mindful of the order).