9 problems

Let $G$ be a finite group of order $n$, let $o(g)$ denote the order of an element $g\in G$, and define … Write $C_n$ for the cyclic group of order $n$, and let $A_5$ denote the alt…

Let $p(G)$ be the smallest value of $\chi(X)-\sigma(X)$ over all closed oriented $4$-manifolds $X$ with $\pi_1(X)=G$, where $\sigma(X)$ denotes the signature of $X$. For the free a…

Solvability conjecture. If

Let $G$ and $H$ be finite abelian groups of the same order. Define the sum-of-element-orders function by … where $o(g)$ denotes the order of $g$. Tărnăuceanu's conjecture. The grou…

Let $G$ be a finite group, and let $G_p$ be a Sylow $p$-subgroup of $G$ for each prime $p$. Define … where $d(G_p)$ is the minimum size of a generating set of $G_p$, and let $m(G)$…

Let $G$ be a finite group, and let $T_G(1,1)$ denote the group invariant introduced by De Medts and Tarnauceanu. De Medts–Tarnauceanu inequality conjecture. For every finite group…

Consider a finite group $G$. Let $m$ denote the minimal dimension of a faithful completely reducible representation of $G$, let $\operatorname{MaxDim}$ denote the maximum size of a…

Let $SK$ and $GK$ denote the square and granny knots, respectively, let $G_n(K)$ be the generalised knot group, and let $H$ be a finite group. Lin–Nelson's homomorphism-counting co…

Let $SK$ and $GK$ denote the square and granny knots, respectively, and let $G_n(K)$ be the generalised knot group associated with a knot $K$. Lin–Nelson's conjecture. The groups ……