110 problems

Non-inner automorphism conjecture. Every finite non-abelian $p$-group has a non-inner automorphism of order $p$.

For a fixed $n\in\mathbb{N}$, let $f_n(p)$ denote the number of isomorphism classes of finite $p$-groups of order $p^n$. Higman's PORC conjecture. There exist a fixed integer $N$ a…

Oliver's p-group conjecture.

Let $G$ be a finite $p$-group, where $p$ is prime. The Schur multiplier exponent conjecture. … This conjecture strengthens the known bound suggested by the counterexamples to Schur…

Mahalanobis's conjecture. Every Miller $p$-group is special for odd $p$.

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

Let $p$ be an odd prime, and let the $p$-class tower group of a field $K$ be the Galois group of its maximal unramified $p$-extension, equivalently the pro-$p$ completion of…

Let $G$ be a finite $p$-group. Write $cl(G)$ for the nilpotency class of $G$ and $b(G)$ for its breadth, and let $Cl(G)$ denote the associated generalized class parameter used in t…

Berkovich's conjecture. Every finite non-abelian $p$-group admits an automorphism of order $p$ which is not inner.

Shank–Wehlau conjecture. The invariant ring $S^G$ is a polynomial ring if and only if $\operatorname{Im}\operatorname{Tr}^G$ is a principal ideal of $S^G$.

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. For an integer $m$, write $m_p$ for its $p$-part, and define … Let $b(P)$ be the largest…

Hughes' conjecture. If $1\ne H_p(G)\ne G$, then

Coexponent conjecture. For each $f$, there is a finite set of primes such that, if $P$ is a finite $p$-group with coexponent $f(P)=f$ and $p$ is not in that set, then

Let $p$ be a prime and $r$ a positive integer. The graph $\mathcal{G}(p,r)$ has finitely many components, and, after removing finitely many sporadic groups, its components are fini…

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…

Let $p$ be a prime and let $G$ be a non-cyclic finite $p$-group of order at least $p^3$. Automorphism-order conjecture. … This conjecture concerns whether every non-cyclic finite…

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.

Ashurst's conjecture. Every fiber of a word map on $G$ satisfies

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

Leedham-Green–Neumann–Wiegold class-breadth conjecture. The nilpotency class of $G$ is at most $b(G)+1$.

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…