8 problems

Let $R$ be a finitely generated commutative ring with unity that is infinitely generated as a $\b7\mathbb{Z}$-module. For $n\b7\geq 2$, let $\b7\mathbf{A}_n(R)$ be the corresponding…

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 $A$ be a finite skew brace with multiplicative group $(A,\cdot)$ and circle group $(A,\circ)$. Byott's conjecture. If $(A,\cdot)$ is soluble, then $(A,\circ)$ is soluble. The p…

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 $G=AB$ 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…

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