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) (number of subgroups)
|
trivial subgroup |
trivial subgroup |
1 |
8 |
1 |
Yes |
1 |
1
|
 |
center of quaternion group |
2 |
4 |
1 |
Yes |
1 |
1
|
 |
cyclic maximal subgroups of quaternion group |
4 |
2 |
1 |
Yes |
2 |
2
|
 |
cyclic maximal subgroups of quaternion group |
4 |
2 |
1 |
Yes |
2 |
2
|
 |
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
-subgroup and the quaternion group, or equivalently the group is a 2-nilpotent group.
Some examples are below:
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) (number of subgroups)
|
trivial subgroup |
trivial subgroup |
1 |
8 |
1 |
Yes |
1 |
1
|
 |
center of quaternion group |
2 |
4 |
1 |
Yes |
1 |
1
|
, ,  |
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: