21 problems

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…

Conjecture W. There are integers $n$ and $k$ such that for all $i\geq n$, the branches satisfy the additional periodicity conditions stated in the source.

Newman–O'Brien's periodicity conjecture. There exist integers $d$ and $n$ such that

The $\mathcal G(5,1)$ periodicity conjecture. If $i\geq 9$, then the branches $\mathcal B_i$ of $\mathcal T$ have depth $i+1$, and

Coclass Conjecture E. For fixed $p$ and $r$, there are only finitely many isomorphism classes of infinite soluble pro-$p$-groups of coclass $r$.

Coclass Conjecture D. For fixed $p$ and $r$, there are only finitely many isomorphism classes of infinite pro-$p$-groups of coclass $r$.

Coclass Conjecture C. Every pro-$p$-group of finite coclass is soluble.

Coclass Conjecture B. For every prime $p$ and every positive integer $r$, there is a positive integer $g$ such that every $p$-group with coclass at most $r$ has soluble length at m…

Let $\ell$ be a positive integer. Theorem, Corollary and Corollary are the results established earlier in the paper for the corresponding construction, with $\ell$ bounded by $8$.…

Let $K$ be a quadratic field, and let $G=\operatorname{Gal}(\mathrm{F}_3^2(K)\mid K)$ be its second $3$-class group. The TKT a.1 defect-zero exclusion conjecture. The transfer kern…

Generalized mainline depth-defect conjecture. The property established for mainline groups of $cmathcal{G}(3,r)$ should also hold for mainline groups on any coclass tree of…

Weak leaf conjecture. Such a vertex $G$ cannot be realized as the second $p$-class group $cmathrm{G}_p^2(K)$ of a quadratic field $K=cmathbb{Q}(csqrt{D})$.

Weak leaf conjecture. Mainline vertices are forbidden as second $3$-class groups $cmathrm{G}_3^2(K)$ of quadratic fields.

Quadratic-field coverage conjecture. The entire tree $cmathcal{T}(V_4)$ is covered by second $2$-class groups $cmathrm{G}_2^2(K)$ of complex quadratic fields…

Mainline depth-defect conjecture. The property described for mainline groups of $cmathcal{G}(p,1)$, namely that their depth is related to their defect through the transfer-kernel b…

Let $p$ be a prime, let $k$ be a field of characteristic $p$, and let $(G_i \mid i \geq 0)$ be a coclass family of finite $p$-groups. Cohomology stabilization conjecture. There exi…

Let $K$ be a field. The graph $\mathcal G_{2,K,2}$ is the subgraph associated with the relevant two-generated nilpotent semigroups of coclass $2$. Five-path conjecture. The graph…

Let $r\in\mathbb N_0$ and let $K$ be an arbitrary field. Let $\mathcal T$ be a maximal coclass tree in $\mathcal G_{r,K}$ with maximal infinite path $A_1\to A_2\to\cdots$. Write…

Let $r\in\mathbb N_0$ and let $K$ be an arbitrary field. For an algebra $A$ in $\mathcal G_{r,K}$, write $\mathcal T(A)$ for the rooted subgraph consisting of all paths starting at…

Let $r\in\mathbb N_0$ and let $K$ be an arbitrary field. The graph $\mathcal G_{r,K}$ has rooted-tree components, and a maximal infinite path is an infinite path starting at the ro…

The paper considers infinite sequences of finite $p$-groups with fixed coclass and observes that their Schur multiplicators can be described by a single parametrised presentation.…