9 problems

Let $K/H$ be the set of equivalence classes of $K$ under the action of $H$, and let $N$ denote the length of a surjective strategy. Orbit-bound conjecture. If $G\wr H$ has a surjec…

Let $G\wr H$ be a wreath product with state group $K$, and call a surjective strategy minimal when it has the form $\{k_i\in K\}_{i=1}^{|K|-1}$. Minimal strategy conjecture. Whenev…

Let $K$ be the state group of a generalized spinning switches puzzle, and measure performance by the expected number of moves when the initial hidden state is chosen uniformly. Asy…

Let $\mathcal{L}=(\{1,a,b,c,d\},\ast)$ be the smallest loop that is not a group, with multiplication operation $\ast$. Loop-switch strategy conjecture. There exists a nontrivial ad…

Let $G\wr H$ be a wreath product, and call a surjective strategy palindromic when its sequence of moves $S'=\{k'_i\in K\}_{i=1}^N$ satisfies $k'_i=k'_{N-i+1}$ for every $i$. Palind…

Let $G$ be a group generated by elements of order $p^n$, let $H$ be a $p$-group acting faithfully on a set of switches $\Omega$, and let $G\wr_\Omega H$ denote the corresponding wr…

Let $A_n$ be the alternating group on $n$ letters, let $C_3$ be the cyclic group of order $3$, and let $A_n\wr C_3$ denote their wreath product. Alternating-group wreath-product st…

Let $G$ be a finite group generated by involutions, and let $H$ be a $2$-group acting faithfully on the set of switches. The wreath product is denoted by $G\wr H$. Prime-power wrea…

Let a finite group be a group with finitely many elements, and call it a 2-group when its order is a power of $2$. Folklore conjecture. Almost all groups are $2$-groups. This conje…