Group acts on set of subgroups by conjugation

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This article describes a pivotal group action: a group action on a set closely associated with the group. This action is important to understand and remember.
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Statement

Let G be a group and S denote the set of subgroups of G. Consider the following action of G on S:

g\cdot H = gHg^{-1}.

In other words, any subgroup is sent to the conjugate subgroup by that group element.

This is a well-defined group action: in other words, gHg^{-1} is a subgroup for every g \in G, and the conditions for a group action are satisfied. Further:

  1. The orbits are the conjugacy classes of subgroups. In other words, two subgroups are conjugate if and only if they are in the same orbit.
  2. The fixed points under the action are precisely the normal subgroups.
  3. The stabilizer (i.e., the isotropy subgroup) of a subgroup H under this action is the normalizer N_G(H): the set of those g \in G for which gHg^{-1} = H.
  4. There is a bijection between the left coset space G/N_G(H) and the set of conjugate subgroups to H. In particular, the number of conjugate subgroups to H in G equals the index of the normalizer N_G(H).

Related facts

Part (4) of the statement tells us that the number of conjugate subgroups of H equals the index of a bigger subgroup, namely, the normalizer of H. Since index is multiplicative, and H \le N_G(H), we have:

[G:N_G(H)][N_G(H):H] = [G:H].

Thus, the number of conjugate subgroups to H is not more than the index of H. In fact, composing the natural maps used in the proofs of these statements yields a map from the coset space of H to the set of conjugate subgroups: the coset gH maps to the conjugate subgroup gHg^{-1}.

Some related terminology:

  • Self-normalizing subgroup is a subgroup that equals its normalizer. In particular, the map sending each coset to the corresponding conjugate subgroup is a bijection.
  • Normal subgroup is a subgroup whose normalizer is the whole group.
  • Almost normal subgroup is a subgroup that has only finitely many conjugate subgroups; equivalently, the normalizer has finite index.

Some further related facts:

  • Union of all conjugates is proper: In a finite group, the union of all conjugates of a proper subgroup is proper. This follows from the fact that the number of conjugate subgroups is not more than the number of cosets, and any two subgroups intersect (at least) at the identity element.

Facts used

  1. Group acts as automorphisms by conjugation: The action of a group G on itself defined by g \cdot x = c_g(x) := gxg^{-1} is a group action, and every element of the group acts as an automorphism.
  2. Group action on set defined group action on power set
  3. Fundamental theorem of group actions: If G acts on a set S and T \subseteq S is the orbit of a point x \in S, then there is a bijection between the cosets of the stabilizer of x and the elements of T.

Proof

Proof that we have a group action

  1. Since the action of G on itself by conjugation is a group action (fact (1)), the induced action on the set of subsets is also a group action (by fact (2)).
  2. Thus, to show that the induced action on the set of subgroups is a group action, it suffices to show that H is a subgroup of G if and only if the conjugate gHg^{-1} is: For this, note that by fact (1), the map c_g = x \mapsto gxg^{-1} is an automorphism. In particular, it sends subgroups to subgroups, and the inverse image of a subgroup is also a subgroup. Thus, H is a subgroup of G if and only if gHg^{-1} is.

Proof of the other consequent facts

  1. Orbits are the conjugacy classes of subgroups: This is direct from the definition.
  2. Fixed points are the normal subgroups: A fixed point under the action is a subgroup H such that gHg^{-1} = H for all g \in G, which is the definition of a normal subgroup.
  3. Stabilizer is the normalizer: This is again direct from the definition.
  4. Bijection between left coset space of normalizer and the set of conjugate subgroups of H: This follows from fact (3), setting the point x as the subgroup H, T as the set of its conjugate subgroups, and noting that the stabilizer of H equals its normalizer.