13 problems

Shi's conjecture. Each finite simple group $G$ is determined by its order $|G|$ and its spectrum $(G)$.

Let $k$ be a fixed positive integer and let $G$ be a finite group. Suppose that at least half of the elements of $G$ have order $k$. Deaconescu's conjecture. Then $G$ is soluble. T…

Relative element-order conjecture. The inequality

Khukhro–Moreto–Zarrin conjecture. If

Let $G$ be a finite group of order $n$ and let $H$ be a subgroup of order $m$ of $G$. For $x\in G$, let $o_H(x)$ be the least positive integer $k$ such that $x^k\in H$, and define…

Let $\operatorname{o}(\operatorname{Sym}(k))$ denote the number of element orders in the symmetric group $\operatorname{Sym}(k)$, and define … for $k\in\mathbb{N}^+$. Maximum-value…

Herzog313131Longobardi313131Maj conjecture. One should have

T3arn3131ceanu's conjecture. If

Supersolvability conjecture. If

Let $G$ and $H$ be finite abelian groups of order $n$. For each $k\in\{1,2,\ldots,n\}$, define … where $G=\{x_1,x_2,\ldots,x_n\}$ and $o(x_i)$ denotes the order of $x_i$. Conjectur…

Zar's conjecture. For all distinct prime divisors $p$ and $q$ of $|G|$, one has

Nilpotent-group conjecture. One has

Let $G$ be a finite group of order $n$, let $C_n$ be the cyclic group of order $n$, and write $o(x)$ for the order of an element $x$. Element-order divisibility matching conjecture…