14 problems

Shareshian's conjecture. Every open interval in the lattice of subgroups of a finite group has the homotopy type of a wedge of spheres.

Let $G$ be an abstract group, and let $\EuScript{L}$ be a lattice of its subgroups. The lattice admits sandwich classification if it is a disjoint union of intervals … where each…

Parzanchevski–Puder free-extension conjecture.

Miasnikov–Ventura–Weil conjecture. For every $H\leqslant_{\mathrm{fg}} F_A$,

Order-preserving subgroup-lattice conjecture. The complex $\Sigma(G_1)$ is isomorphic to $\Sigma(G_2)$ if and only if there exists an isomorphism between their subgroup lattices th…

Let $G$ be a cyclic group whose order has three distinct prime factors, and let $\Gamma(G)$ denote its subgroup lattice graph. Proposition upper bounds the genus…

Let $H\leqslant_{\mathrm{fg}}\Free[A]$, where $A$ is a basis of the free group $\Free[A]$. For each ambient basis $A'$ of $\Free[A]$, let $\mathcal{O}_{A'}(H)$ denote the correspon…

Parzanchevski–Puder conjecture. For every $H\leqslant_{\mathrm{fg}} F_A$,

Qu's conjecture. The inequalities

Let $G$ be a finite group, and consider an open interval in the subgroup lattice of $G$. Shareshian's conjecture. Every such open interval has the same homotopy type as a wedge of…

Let $[H,G]$ be an interval of finite groups. Its dual Euler totient is … An interval $[H,G]$ is Boolean when its subgroup-lattice interval is a Boolean lattice. Nonvanishing dual E…

Shareshian's conjecture. The displayed lattice is not realizable as an intermediate subgroup lattice $[H,G]$ of any finite group.

Let $S_n$ be the symmetric group of degree $n$, and let $\mu(1,S_n)$ denote the Möbius number of the subgroup lattice of $S_n$. Symmetric-group Möbius number conjecture. For all…

Let $G$ be a finite solvable group, and let $L(G)$ denote its subgroup lattice. Choose a chief series … A complement to a subgroup $N$ is a subgroup $H$ with $HN=G$ and $H\cap N=1$…