Fusion systems for quaternion group

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

This article discusses possible fusion systems for the quaternion group.

Summary

Item	Value
Total number of saturated fusion systems on a concrete instance of the group (strict, not up to isomorphism of fusion systems)	2
Total number of saturated fusion systems up to isomorphism	2
List of saturated fusion systems with orbit sizes	inner fusion system (orbit size 1 under isomorphisms), non-inner fusion system for quaternion group (orbit size 1 under isomorphisms)
Number of simple fusion systems	None (the reason is that in every fusion system, the fusion subsystem induced on the center of quaternion group is normal)
Number of maximal fusion systems, i.e., fusion systems not contained in bigger fusion systems	1 (the non-inner case)

Description of saturated fusion systems

Isomorphism type of fusion system	Number of such fusion systems under strict counting	Can the fusion system be realized using a Sylow subgroup of a finite group?	Does the identity functor control strong fusion? This would mean that all fusion occurs in the normalizer	Is the fusion system simple?	Smallest size embedding realizing this fusion system (if any)
inner fusion system	1	Yes	Yes	No	as a subgroup of itself
non-inner fusion system for quaternion group	1	Yes	Yes	No	Q8 in SL(2,3)

Inner fusion system

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	trivial subgroup	1	8	1	Yes	1	1
$\{1,-1\}$	center of quaternion group	2	4	1	Yes	1	1
$\{1,-1,i,-i\}$	cyclic maximal subgroups of quaternion group	4	2	1	Yes	2	2
$\{1,-1,j,-j\}$	cyclic maximal subgroups of quaternion group	4	2	1	Yes	2	2
$\{1,-1,k,-k\}$	cyclic maximal subgroups of quaternion group	4	2	1	Yes	2	2
whole group	whole group	8	1	1	No	4	4

Sylow subgroups realizing this fusion system

This fusion system is realized by a group having quaternion group 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 quaternion 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 Q8 and Z3	24	cyclic group:Z3	Yes
dicyclic group:Dic24	24	cyclic group:Z3	No
direct product of Q8 and Z5	40	cyclic group:Z5	Yes
dicyclic group:Dic40	40	cyclic group:Z5	No

Fusion system using an outer automorphism of order three

Equivalence class under isomorphisms	Subgroups involved	Order	Index	Number of subgroups (=1 iff weakly closed subgroup)	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	trivial subgroup	1	8	1	Yes	1	1
$\{1,-1\}$	center of quaternion group	2	4	1	Yes	1	1
$\{1,-1,i,-i\}$, $\{1,-1,j,-j\}$, $\{1,-1,k,-k\}$	cyclic maximal subgroups of quaternion group	4	2	3	Yes	2	18
whole group	whole group	8	1	1	Yes	24	24

Sylow subgroups realizing this fusion system

Any situation where quaternion group arises as a 2-Sylow subgroup that is not a retract, i.e., does not have a normal complement. Any such example must admit special linear group:SL(2,3) as a subquotient. Examples are given below:

Group	Order	Quaternion group as a subgroup of this group	Comment
special linear group:SL(2,3)	24	Q8 in SL(2,3)
special linear group:SL(2,5)	120	Q8 in SL(2,5)	There is an intermediate SL(2,3) in SL(2,5) that controls all the fusion behavior.