Fusion systems for dihedral group:D8

This article discusses the possible fusion systems for the dihedral group of order eight.

$$G = \langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

There are, up to isomorphism, three possible fusion systems on $$G$$.

The inner fusion system: the fusion system obtained from inner automorphisms
This is the fusion system where all morphisms are obtained as the restriction of inner automorphisms of $$G$$. The isomorphisms are as follows.

Sylow subgroups realizing this fusion system
This fusion system is realized by a group having dihedral group:D8 as its 2-Sylow subgroup if and if it possesses a normal complement, so the 2-Sylow subgroup is a retract of the group and the group is a semidirect product of a normal $$p'$$-subgroup and the dihedral group, or equivalently the group is a 2-nilpotent group.

Some examples are below:

Simple fusion system
Below is a description of the unique simple fusion system on this group.