Proving that a subgroup is conjugacy-closed
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This article is about strategies to prove that a subgroup (either a chosen one, or one subject to certain constraints) is conjugacy-closed in the whole group.
The definition of conjuacy-closed subgroup
A subgroup of a group is termed self-centralizing in , if no two distinct conjugacy classes of get fused in . Equivalently, if are such that , then there exists such that .
Typical strategies used
When both groups are well-understood
When both and are well-understood groups, we use what we know about conjugacy classes in and in . In other words:
- We determine a general criterion for elements of to be conjugate, and interpret this criterion when the two elements both happen to be from .
- We determine a general criterion for elements of to be conjugate.
- We then check that the two criteria are equal.
This usually involves a good understanding of both groups, which may or may not be available.
Finding a conjugate-dense subgroup, or subset
This is a strategy that works well, specially when the conjugacy classes in are well-understood. First, we find a subgroup of satisfying the following two conditions:
- is conjugate-dense in : in other words, every element of is conjugate in , to some element in .
- Any two elements of that are conjugate in are conjugate in . In other words, the fusion of in is completely controlled by .
This strategy works well when proving results about linear groups, because in these cases, the conjugacy classes are well-understood. In fact, we don't even need to be a subgroup above. All we need is a subset of satisfying:
- Every element of is conjugate to some element of
- Any two elements of that are conjugate in are also conjugate in
Here are some situations where we can apply this strategy:
- General linear group over subfield is conjugacy-closed: In this case, we use the rational canonical form for the general linear group over the subfield, and argue that if two matrices in the rational canonical form over the subfield are conjugate in the big field, they're also conjugate in the subfield.
- Unitary group is conjugacy-closed in general linear group: This states that the group of unitary matrices over is conjugacy-closed inside the group . For this, we use the spectral form theorem to first argue that any unitary matrix is conjugate, in , to a diagonal unitary matrix. Thus, the subgroup of diagonal unitary matrices is conjugate-dense in the group . Next, we show that any two diagonal unitary matrices that are conjugate in have the same entries upto permutation, hence they are conjugate by a permutation matrix, and hence they are conjugate in .