10 problems

Locally finite-group Addition Theorem conjecture. If $G$ is a locally finite group, then $AT(G)$ holds.

Let $G$ be a locally finite group of finite $c$-dimension $k$. Let $F(G)$ denote the Hirsh–Plotkin radical of $G$, and let $F^*(G/F(G))$ denote the generalized Fitting subgroup of…

Let $G$ be a locally finite group, let $R$ be a nonzero $o(G)$-inverting ring, and set … The cohomological dimension conjecture asserts that … Equivalently, after reducing to finit…

Existence conjecture. There are $p'$-reduced locally finite $p$-artinian groups that are not locally finite, strongly $p$-artinian groups, and there are fusion systems that are LF-…

Let $n>3$ and let a lattice mean a discrete subgroup of $\operatorname{Isom}(\mathbb{H}^n)$ with finite covolume. A subgroup is LERF if every finitely generated subgroup is an inte…

Let $\mathbb{U}$ be the Urysohn space, and let $\mathbb{H}$ be Hall's universal countable locally finite group. The group $\operatorname{Iso}(\mathbb{U})$ is the isometry group of…

Bounded-commutator local finiteness conjecture. The derived subgroup $G'$ is locally finite.

Abelian p-group reduction conjecture. $Aut_H(G)$ has a locally finite subgroup of power $\lambda$.

Automorphism-group conjecture. If $G \in \mathbf{K}^{\lambda}_{\mathrm{lf}}$, then $Aut(G)$ has a locally finite subgroup of power $\lambda$.

Borovik's conjecture. The number of nonabelian composition factors of $G$ is bounded in terms of $k$.