49 problems

Let $G$ be a finite $p$-group, let $k$ be a positive integer, and let $d$ and $a_1,\dots,a_d$ be integers. A regular sequence has degree sequence $a_1,\dots,a_d$ when its elements…

Let $G$ be a finite $p$-group. Its cohomology ring $H^*(G,\mathbb F_p)$ is a graded-commutative, noetherian, local $\mathbb F_p$-algebra. Write $\operatorname{rank}_p Z(G)$ for the…

Miller's conjecture. If $G$ is a Miller $p$-group for an odd prime $p$, then $G$ is special.

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, and let $\chi, \psi \in \operatorname{Irr}(G)$ be faithful characters. Let $\eta(\chi,\psi)$ denote the number of d…

Medvedev's conjecture. $L$ contains a solvable subring of $m$-bounded derived length whose index is $(p,n,m)$-bounded.

Khukhro's conjecture. $G$ contains a subgroup of $m$-bounded derived length and $(p,n,m)$-bounded index.

Khukhro's conjecture. $G$ contains a subgroup of $m$-bounded nilpotency class and $(p,m)$-bounded index.

Strengthened class-breadth conjecture. There exists a $G$-central series $C_i$, $0\leq i\leq N$, of $P$ such that

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Isaacs–Navarro–Tiep conjecture. Then … This strengthens one of the preceding conjectures…

Sibling-pair and sibling-tuple conjecture. Among the $15$ groups of order $p^4$, there is exactly one sibling-pair. Among the $2p+2\gcd(p-1,3)+\gcd(p-1,4)+61$ groups of order $p^5$…

Twig periodicity conjecture. There exist integers $e=e(p)$ and $f=f(p)$ with $(p-1)\mid f$ such that, for every $i\geq e$ and every $\gamma\in\hat{H}_i$,

Terminal-leaf conjecture. The leaves of the frame $\mathcal F_i$ are terminal groups in $\mathcal B_i$.

Nontriviality conjecture. For every such $p$, $i$, and $\gamma$,

Let $G$ be a finite $p$-group. Denote by $\gamma_i(G)$ the $i$th term of the lower central series of $G$, by $Cl(G)$ the generalized class parameter used in the paper, and by…

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-…

Oliver's p-group conjecture.

Let $p\geq 5$ be prime, and let an exceptional group be a finite group $G$ having a normal subgroup $N$ such that $\mu(G/N)>\mu(G)$, where $\mu(G)$ is the least integer $n$ for whi…

Green–Héthelyi–Lilienthal reformulation. If $V$ is an $F$-module, then there is an element $1\ne g\in\Omega_1(Z(G))$ such that the minimal polynomial of the action of $g$ on $V$ d…

Let $G$ be a finite $p$-group, let $E$ be a $p$-complete complex-oriented spectrum with associated formal group of height $n$, and let $E^2(f)$ denote the induced map in degree-two…

Minimal-zero conjecture.

The zeros conjecture. $G$ has an irreducible character that vanishes at exactly $3^n-3^{n-1}+6$ elements if and only if $G$ is a $3$-group of maximal class with a maximal abelian s…

Let $p$ be a prime, let $G$ be a finite group whose order is divisible by $p$, and let $P\in\operatorname{Syl}_p(G)$. p-local lower-bound conjecture. The number of irreducible $p'$…

Let $G$ be a finitely generated pro-$2$ group, and write $d(G)=\dim_{\mathbb{F}_2}G/G^{2}[G,G]$. Call $G$ $d$-maximal if $d(H)<d(G)$ for every proper closed subgroup $H<G$. The pro…

Let $r\geq 1$ be an integer. A finite $2$-group $G$ is $d$-maximal if $d(H)<d(G)$ for every proper subgroup $H<G$, where $d(G)$ is the minimal number of generators. The bounded-cla…

Throughout, let $G$ be a finite $2$-group, let $d(G)$ denote its minimal number of generators, and let $r_G$ be the real regular representation of $G$. Define … where $w_i(r_G)$ is…