Fischer graph of a subgroup

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Definition

Definition with symbols

Let A be a subgroup of a group G, and let S be the set of all conjugates of A. Then the Fischer graph \Gamma associated with A is defined as follows:

  • Its vertex set is S (viz, the vertices are conjugates of A)
  • Two vertices in S are adjacent if they commute element-wise. Viz, A and B are adjacent if [A,B] = 1

The group G acts on the Fischer graph \Gamma by conjugation.

Facts

Connected complement

The graph-theoretic complement of the Fischer graph of a contranormal subgroup is connected.