Proving that a subgroup is conjugate-dense
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This article discusses general strategies for proving that a subgroup of a group is a conjugate-dense subgroup, i.e., that every element of the whole group is conjugate to some element of the subgroup.
Note that for a finite group, no proper subgroup can be conjugate-dense. More generally, in any group, no proper subgroup of finite index can be conjugate-dense. Further information: Union of all conjugates is proper
Most of the strategies discussed here work not just for subgroups, but for arbitrary subsets. In other words, given a group and a subset
, these strategies help prove that every element of
is conjugate to some element of
. While the most special case is that where
is a subgroup of
, other cases of interest arise, for example, when
is a union of a few well-chosen subgroups.
Related techniques
The general strategy
Suppose is a group and
is a subgroup (or more generally, subset) of
. Then,
acts on itself by conjugation. We want to show that every element of
is in the orbit of some element of
. Equivalently, we want to show that starting with any element
, we can find an element
such that
.
The step-by-step approach
In this approach, we think of the elements of as extreme elements, and create a gradation in the elements of
. Next, we show that, starting with any arbitrary element
,