The exceptional Klein four group

Question

There are at least four major cases I know where $V\cong C_2\times C_2$ is an exceptional group:

• The commuting probability of a nonabelian group $G$ (the probability two elements drawn uniformly at random from $G$ commute) is maximized precisely when $G/Z(G)\cong V$.
• A group $G$ cannot be the union of two proper subgroups, however Scorza's theorem implies that $G=H_1\cup H_2\cup H_3\iff G/(H_1\cap H_2\cap H_3)\cong V$ for proper $H\subset G$.
• The only vector space canonically isomorphic to its dual is $V$. See Martin's answer here, the idea traces back to ACL's comment which gives earlier attribution. Given $\{0,a,b,c\}$ is a copy of $V$, one can define e.g. the dual vector $a^*$ to be the characteristic function of $\{b,c\}$.
• Say $G$ is a functor from the category ${\sf B}_n$ of sets of cardinality $n$ with bijections into $\sf Grp$, equipped with a natural transformation $G\to{\rm Perm}$ consisting of injective group homomorphisms. One calls $GX$ a natural permutation group on $X$. The only natural permutation groups are trivial, alternating, symmetric, and $V$ on four element sets.

The last is of my own making and follows from $V\triangleleft S_4$ being the only exceptional normal subgroup of symmetric groups (arguably I should have just stated that as my bullet point). It is the unique subgroup of ${\rm Perm}(\{a,b,c,d\})$ which fixes every partition of the form $\{\{\alpha,\beta\},\{\gamma,\delta\}\}$, and thus can be specified canonically without making any arbitrary choices. I also have a vague sense that $V$s automorphisms ${\rm Aut}(V)\cong S_3$ act exceptionally transitively, but that's likely just a special case of the same phenomenon for all elementary abelian groups.

Questions:

1. Are the above examples related to each other, or is there a unifying explanation for why we should expect $V$ to manifest as an edge case in so many different guises?
2. Does anybody have more examples of $V$ being an exceptional group to add to the list?

Answer 1

2. For any group $G$, $Aut(G)$ is clearly a subgroup of the permutations of the set of non-trivial elements of $G-\{1_G\}$. One can prove that $Aut(G)=S_{G-\{1_G\}}$ if and only if $G$ is the Klein group. In this sense the Klein group is the unique group with a "maximal" automorphism group.

$$|Aut(G)|\leq (|G|-1)!\text{ with equality iff } G\text{ is isomorphic to the Klein group.}$$

So to end your remark $V$ acts exceptionally transitively but this is not a special case for elementary abelian group.