Fusion systems for dihedral group:D8

This article gives specific information, namely, fusion systems, about a particular group, namely: dihedral group:D8.
View fusion systems for particular groups | View other specific information about dihedral group:D8

There are multiple conventions on whether the term fusion system should refer to saturated fusion system or to a weaker notion. For our purposes, we will mostly be interested in saturated fusion systems, because these mimic/generalize situations where the

$p$

-group is a

$p$

-Sylow subgroup of some finite group. Unless otherwise specified, we are referring to saturated fusion systems when we talk about fusion systems.

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, three possible fusion systems on $G$.

Family contexts

Family name	Parameter values	General information on fusion systems for family
dihedral group (of degree, order powers of 2)	degree $n=4$, order $2n=8$	fusion systems for dihedral groups

For more on comparison with other groups of order 8, see fusion systems for groups of order 8.

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) $\times$ ${\text{(number of subgroups)}}^2$
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

Fusion system where the center is identified with one other conjugacy class

Explicit description of fusion system

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 (one of the two conjugacy classes)	2	4	3	Yes	1	9
$\{e,x\},\{e,a^{2}x\}$	$\{(),(1,3)\},\{(),(2,4)\}$	non-normal subgroups of dihedral group:D8 (one of the two conjugacy classes)	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

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 fusion system for this is precisely the same as that for D8 in S4 viewing symmetric group:S4 as an intermediate subgroup.

Simple fusion system

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

Explicit description of fusion system

Equivalence class under isomorphisms, explicit description of subgroups	Subgroups involved	Order	Index	Number of conjugacy classes of subgroups fused	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) $\times$ ${\text{(number of subgroups)}}^2$
trivial subgroup		1	8	1	1	Yes	1	1
$\{e,a^2\}$ $\{e,x\},\{e,a^{2}x\}$ $\{e,ax\},\{e,a^{3}x\}$,	center of dihedral group:D8, non-normal subgroups of dihedral group:D8 (both conjugacy classes)	2	4	3	5	Yes	1	25
$\{e,a,a^2,a^3\}$	cyclic maximal subgroup of dihedral group:D8	4	2	1	1	Yes	2	2
$\{e,x,a^2,a^{2}x\}$	Klein four-subgroups of dihedral group:D8 (one of them)	4	2	1	1	Yes	6	6
$\{e,ax,a^2,a^{3}x\}$	Klein four-subgroups of dihedral group:D8 (one of them)	4	2	1	1	Yes	6	6
whole group		8	1	1	1	No (see below)	4	4
Total		--	--	8	10	--	--	--

For the whole group, the only automorphisms that are included in the fusion system are the inner automorphisms. This can be traced to the fact that the outer automorphism group of the dihedral group is 2-group, and in particular has no 2'-automorphisms.