Focal subgroup of a subgroup

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Definition

Let H be a subgroup of a group G. We define the focal subgroup of H in the following equivalent ways:

  • The subgroup generated by the left quotients of pairs of elements of H which are conjugate in G.
  • The subgroup generated by the right quotients of pairs of elements of H which are conjugate in G.

We use the notation \operatorname{Foc}_G(H) or H^*_G for the focal subgroup of H in G.

Note that the focal subgroup of H in G is contained within the commutator [H,G], and contains the commutator [H,H]. In fact, we have the following string of inequalities:

[H,H] \le \operatorname{Foc}_G(H) \le H \cap [H,G] \le H \cap [G,G].

Facts