Fusion systems for dihedral group:D8

This article gives specific information, namely, fusion systems, about a particular group, namely: dihedral group:D8.
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This article discusses the possible fusion systems for the dihedral group of order eight.

$G=⟨a,x\mid a^4=x^2=e,xa{x}^{-1}=a^{-1}⟩$.

There are, up to isomorphism, two 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.

Equivalence class under isomorphisms	Subgroups involved	Order	Index	Total number of subgroups	Are all group automorphisms of each subgroup included?	Size of automorphism group from the fusion system	Total number of isomorphisms (including automorphisms and others) = (number of automorphisms) $\times$ (number of subgroups)$2^{}$
trivial subgroup		1	8	1	Yes	1	1
$\{e,a^2\}$	center of dihedral group:D8	2	4	1	Yes	1	1
$\{e,x\},\{e,a^{2}x\}$	non-normal subgroups of dihedral group:D8 (some of them)	2	4	2	Yes	1	4
$\{e,ax\},\{e,a^{3}x\}$	non-normal subgroups of dihedral group:D8 (some of them)	2	4	2	Yes	1	4
$\{e,a,a^2,a^3\}$	cyclic maximal subgroup of dihedral group:D8	4	2	1	Yes	2	2
$\{e,x,a^2,a^{2}x\}$	Klein four-subgroups of dihedral group:D8 (one of them)	4	2	1	No	2	2
$\{e,ax,a^2,a^{3}x\}$	Klein four-subgroups of dihedral group:D8 (one of them)	4	2	1	No	2	2
whole group		8	1	1	No	4	4

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 $p^{\prime }$-subgroup and the dihedral group, or equivalently the group is a 2-nilpotent group.

Some examples are below:

Group	Order	Isomorphism class of normal complement	Is it a direct product?
direct product of D8 and Z3	24	cyclic group:Z3	Yes
dihedral group:D24	24	cyclic group:Z3	No
SmallGroup(24,8)	24	cyclic group:Z3	No
direct product of D8 and Z5	40	cyclic group:Z5	Yes
dihedral group:D40	40	cyclic group:Z5	No

The other fusion system

This is unique up to automorphisms of the group. There are in fact two versions of this viewed strictly, which are interchanged under the outer automorphism sending $x$ to $ax$ and fixing $a$.

This fusion system is realized, for instance, in the symmetric group of degree four. In fact, the symmetric group of degree four is in essence the only example -- any example admits this as a subquotient.

Equivalence class under isomorphisms, explicit description of subgroups	Description of subgroups as permutation groups	Subgroups involved	Order	Index	Total number of subgroups (=1 iff weakly closed subgroup for the fusion system)	Are all group automorphisms of each subgroup included?	Size of automorphism group from the fusion system	Total number of isomorphisms (including automorphisms and others) = (number of automorphisms) $\times$ (number of subgroups)$2^{}$
trivial subgroup	$()$		1	8	1	Yes	1	1
$\{e,a^2\},\{e,ax\},\{e,a^{3}x\}$	$\{(),(1,3)(2,4)\}$, $(),(1,4)(2,3)\}$, $\{(),(1,2)(3,4)\}$	center of dihedral group:D8, non-normal subgroups of dihedral group:D8 (some of them)	2	4	3	Yes	1	9
$\{e,x\},\{e,a^{2}x\}$	$\{(),(1,3)\},\{(),(2,4)\}$	non-normal subgroups of dihedral group:D8 (some of them)	2	4	2	Yes	1	4
$\{e,a,a^2,a^3\}$	$\{(),(1,2,3,4),(1,3)(2,4),(1,4,3,2)\}$	cyclic maximal subgroup of dihedral group:D8	4	2	1	Yes	2	2
$\{e,x,a^2,a^{2}x\}$	$\{(),(1,3),(1,3)(2,4),(2,4)\}$	Klein four-subgroups of dihedral group:D8 (one of them)	4	2	1	No	2	2
$\{e,ax,a^2,a^{3}x\}$	$\{(),(1,4)(2,3),(1,3)(2,4),(1,2)(3,4)\}$	Klein four-subgroups of dihedral group:D8 (one of them)	4	2	1	Yes	6	6
whole group			8	1	1	No	4	4

The other version interchanges the roles of the two Klein four-subgroups, and also of the two conjugacy classes of non-normal subgroups of order two.

Sylow subgroups realizing this fusion system

Any situation where dihedral group:D8 arises as a Sylow subgroup but is not a retract, i.e., any situation where it does not have a normal complement, it must admit this fusion system (or its equivalent under the outer automorphism). Moreover, in all cases, we can recover a subquotient isomorphic to symmetric group:S4.

Here are some examples:

Group	Order	Dihedral group:D8 as a subgroup in this group	Comment
symmetric group:S4	24	D8 in S4	this is the canonical, minimal example
symmetric group:S5	120	D8 in S5	The intermediate subgroup S4 in S5, and the embedding D8 in S4, have complete control over the fusion behavior.
projective special linear group:PSL(3,2)	168	D8 in PSL(3,2)	The intermediate subgroup S4 in PSL(3,2) has complete control over the fusion behavior.