Proving that a subgroup is conjugate-dense

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This is a survey article related to:conjugate-dense subgroup
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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 G and a subset H, these strategies help prove that every element of G is conjugate to some element of H. While the most special case is that where H is a subgroup of G, other cases of interest arise, for example, when H is a union of a few well-chosen subgroups.

Related techniques

The general strategy

Suppose G is a group and H is a subgroup (or more generally, subset) of G. Then, G acts on itself by conjugation. We want to show that every element of G is in the orbit of some element of H. Equivalently, we want to show that starting with any element g \in G, we can find an element a \in G such that aga^{-1} \in H.

The step-by-step approach

In this approach, we think of the elements of H as extreme elements, and create a gradation in the elements of G. Next, we show that, starting with any arbitrary element g \in G,