Simple fusion system for Klein fourgroup
This article describes a particular fusion system on a group of prime power order, namely Klein fourgroup.
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This article describes the unique noninner saturated fusion system on the Klein fourgroup. This is the unique maximal fusion system on the group. It is also a simple fusion system.
We use to denote the elements of the Klein fourgroup. denotes the identity element and denote the nonidentity elements.
Explicit description
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) 

trivial subgroup  1  4  1  1  Yes  1  1  

Z2 in V4  2  2  3  3  Yes  1  9 
whole group  4  1  1  1  No  3  3  
Total (3 rows)  5  5       
Using endomorphism structure of Klein fourgroup, we know that the automorphism group of the whole group is which is isomorphic to symmetric group:S3. The subgroup of this that is realized in the fusion system is the subgroup A3 in S3. Note that this is the largest subgroup that can be realized, because the Sylow axiom for saturated fusion systems forces us to choose a subgroup of the automorphism group that has order relatively prime to 2.