Simple fusion system for Klein four-group

This article describes a particular fusion system on a group of prime power order, namely Klein four-group.
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This article describes the unique non-inner saturated fusion system on the Klein four-group. This is the unique maximal fusion system on the group. It is also a simple fusion system.

We use $e,a,b,c$ to denote the elements of the Klein four-group. $e$ denotes the identity element and $a,b,c$ denote the non-identity 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) $\times$ $\mbox{(number of subgroups)}^2$
trivial subgroup 1 4 1 1 Yes 1 1 $\{ e, a \}$ $\{ e,b \}$ $\{ e,c \}$
Z2 in V4 2 2 3 3 Yes 1 9
whole group $\{ e,a,b,c \}$ 4 1 1 1 No 3 3
Total (3 rows) 5 5 -- -- --

Using endomorphism structure of Klein four-group, we know that the automorphism group of the whole group is $GL(2,2)$ 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.