16 problems

Let $G$ be a group, let $x\in G$, and write $\operatorname{Sol}_G(x)$ for the set of elements $y\in G$ such that $\langle x,y\rangle$ is soluble. Let $N_G(\langle x\rangle)$ denote…

Soluble poly-context-free group conjecture. Every finitely generated soluble group is poly-context-free if and only if it is virtually abelian.

Let $G$ be a primitive soluble permutation group, and let $b(G)$ denote its base size. A greedy base is a base produced by repeatedly choosing a point in a longest orbit of the cur…

Let $G$ be a finite group and $x\in G$. Define the solubilizer probability of $x$ by … where $\operatorname{Sol}_G(x)$ is the set of elements $y\in G$ such that…

Let $G$ be a finite insoluble group, let $x\in G$, and write $\operatorname{Sol}_G(x)$ for the set of elements $y\in G$ such that $\langle x,y\rangle$ is soluble. Mousavi's odd-pri…

Let $\Gamma$ be a finitely generated residually finite soluble minimax group, and let $M$ be a quasi-finitely generated $\Gamma$-module, meaning that some finitely generated group…

For a finite group $G$, let $\rho_p(G)$ denote its $p$-rank and let $\rho(G)$ denote its rank. For an arbitrary group $G$, let $\mathcal{F}(G)$ be the set of its finite quotient gr…

Let $G$ be a finitely generated soluble group. Its subgroup growth is said to have a subgroup growth gap if there is no subgroup growth rate strictly between polynomial growth and…

Let $G$ be a finitely presented soluble group, and define its first virtual rational Betti number by … where $\mathcal{A}_G$ is the set of all subgroups of finite index in $G$. Bri…

Let $G$ be a finitely generated group of rank $n$. An ordered $n$-tuple is normally generating when its components normally generate $G$, and its recalcitrance is the least number…

Let $G$ be a finitely generated group, let $w(G)$ be its weight, and let $G_{ab}=G/[G,G]$ be its abelianization. For $n\geq\max(w(G),2)$, an ordered $n$-tuple is normally generatin…

Let $G$ be a finitely generated soluble group, and let $Rat(G)$ denote the family of all rational subsets of $G$. Assume that $Rat(G)$ is a boolean algebra. Rational-subset conject…

Brin group embedding conjecture. Every finitely generated non-soluble subgroup $G<PL_+(I)$ contains an embedded copy of Brin's group $B$.

Finiteness conjecture. If $G$ is finitely presented and soluble, then

Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. Here $h(X)$ denotes the Fitting height of a soluble group $X$, and…

Let $G$ be a soluble group of finite Prüfer rank and of type $\operatorname{FP}_n$. Choose an extension $N \rightarrowtail G \twoheadrightarrow Q$, where $N$ is nilpotent and $Q$ i…