Conjugate-dense subgroup
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This is an opposite of normality
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This article is about a general term. A list of important particular cases (instances) is available at Category:Instances of conjugate-dense subgroups
Definition
Symbol-free definition
A subgroup of a group is said to be conjugate-dense if it satisfies the following equivalent conditions:
- The union of all conjugates of the subgroup in the group, is the whole group
- Every element in the whole group is conjugate to some element in the subgroup
- Every cyclic subgroup of the whole group is conjugate to a cyclic subgroup of the given subgroup. In other words, it dominates all cyclic subgroups
- For the action of the whole group on the coset space, every element of the group has a fixed point.
Definition with symbols
A subgroup of a group
is termed conjugate-dense in
if it satisfies the following equivalent conditions:
-
- For any
, there exists
such that
- For every cyclic subgroup
, there exists
such that
- Under the natural action of
on the coset space
, every
has a fixed point.
Relation with other properties
Weaker properties
- Contranormal subgroup: Here, we only require that the normal closure should be the whole group. Note that the normal closure may, in general be much bigger than the union of conjugates.
Incomparable properties
Opposite properties
- Subgroup of finite index: It turns out that there cannot be a conjugate-dense subgroup of finite index other than the whole group. For full proof, refer: Union of all conjugates is proper
- Normal subgroup: Clearly, the only normal conjugate-dense subgroup is the whole group
Importance
The general context in which being conjugate-dense is important is as follows. Suppose is a set with some additional structure, and
is the group of automorphisms with that additional structure. This
could in principle be very huge, and unmanageable. Now suppose adding some further structure to
causes the automorphism group to reduce to a much smaller subgroup
of
.
In principle we could lose a lot of the symmetry in when we pass to
. Thus, we are often interested in the question: when can we guarantee that every
is conjugate (in
) to some element of
? In other words, is
conjugate-dense in
? If the answer to this question is yes, then that means that at least if we are looking at only one element of
at a time, then we might safely assume that our element is in
.
Examples of passage to additional structure are:
- Giving a differential structure to a topological manifold
- Giving a Riemannian structure to a topological or differential manifold (typically, the isometry group of a Riemannian manifold is far from conjugate-dense in the group of diffeomorphisms or homeomorphisms)
- Giving a linear or piecewise linear structure (or a simplicial structure) to a topological manifold
The importance of finite-dominating subgroups is for similar reasons.
Examples
For a full list of examples, refer:
Category:Instances of conjugate-dense subgroups
- The subgroup of upper triangular matrices, viz the Borel subgroup
, is conjugate-dense in
: in other words, every invertible matrix is conjugate to an upper triangular matrix (the analogous result does not hold for a non-algebraically closed field, and in particular, it doesn't hold for any finite field). Further information: Triangulability theorem
- The statement that every element of
is a rotation about some axis (called Euler's theorem) can be rephrased as saying that
is conjugate-dense in
Further information: Euler's theorem
Metaproperties
Transitivity
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
If are subgroups such that
is the union of conjugates of
within
, and
is the union of conjugates of
within
, then:
Every conjugate of within
is expressible as a union of conjugates of
within
.
This forces to be conjugate-dense in
. For full proof, refer: Conjugate-denseness is transitive
Trimness
The property of being conjugate-dense can be satisfied by the trivial subgroup only if the whole group is trivial. However, it is always satisfied by the whole group, hence it is identity-true subgroup property.
Intermediate subgroup condition
The property of being conjugate-dense probably does not satisfy the intermediate subgroup condition.