# Fischer graph of a subgroup

From Groupprops

## Definition

### Definition with symbols

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

- Its vertex set is (viz, the vertices are conjugates of )
- Two vertices in are adjacent if they commute element-wise. Viz, and are adjacent if

The group acts on the Fischer graph by conjugation.

## Facts

### Connected complement

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