Recovering a group from few orbits

Abstract

For an unknown finite group $G$ of automorphisms of a finite-dimensional Hilbert space, we find sharp bounds on the number of generic $G$-orbits needed to recover $G$ up to group isomorphism, as well as the number needed to recover $G$ as a concrete set of automorphisms.

Figures (3)

• Figure 1: Six orbits in ${\mathbb R}^2$ arising from the actions of six different subgroups of the orthogonal group $\operatorname{O}(2)$. The reader is invited to guess the isomorphism class of the group that generated each orbit. In each case, you may assume that the point that generated the orbit was drawn at random according to a continuous probability distribution over $\mathbb R^2$. The solutions can be found in the footnote on the next page.
• Figure 2: Orbits generated by $C_8$ (left) and $D_4$ (right). As discussed in Example \ref{['ex:davinci-actions']}, the points fall on the vertices of a centered regular octagon and a centered truncated square, respectively, each shown in blue.
• Figure 3: An illustration of the orbit pairing lemma (Lemma \ref{['lem:pairing']}) and its proof. (left) A pair of orbits in ${\mathbb R}^2$ generated by a subgroup $C_4\leq \operatorname{O}$(2). The permutation on the inner orbit induced by a generator of $C_4$ is depicted with dashed arrows. (right) Black line segments connect each point in the inner orbit with the nearest point in the outer orbit, representing the equivariant bijection $\beta$ constructed in the proof of Lemma \ref{['lem:pairing']}. The equivariance of this bijection determines the action of the $C_4$ generator on the outer orbit as well, represented with dashed arrows.

Theorems & Definitions (29)

• Example 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Example 2.4
• Theorem 2.5
• proof
• Theorem 2.6: One-orbit theorem
• proof
• Conjecture 2.7