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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?
Chain Markov
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anon
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  • $V$ is the only abelian dihedral group...(unless you count the trivial group, which some do not, as $1$ is not an even number). – David Wheeler Sep 13 '15 at 16:59

1 Answers1

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  1. 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.

Clément Guérin
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    The cyclic groups of order $2$ and $3$ also has the full symmetric group as automorphism groups, but these are the only other such examples. In general, the group of automorphisms is transitive iff the group is an elementary abelian $p$-group, doubly transitive iff $p=2$ and never $3$-transitive. – Tobias Kildetoft Sep 14 '15 at 07:44
  • @TobiasKildetoft, thanks for the correction, I totally forgot about those cases. – Clément Guérin Sep 14 '15 at 07:48