5 problems

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…

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

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 $n$ be a positive integer with $n\geqslant 4$, let $p\in\{2,3\}$, and let $G$ be a $p$-group in which every proper subgroup has nilpotency class at most $n$. Write $c(G)$ for t…