5 problems

Let $G$ be a finite group and let an $S$-ring over $G$ be a subring of $\mathbb{Z}G$ that is a free $\mathbb{Z}$-module spanned by a partition of $G$ closed under taking inverses a…

Let $G$ be a finite cyclic group. A group is called Schur if every S-ring over it is schurian, meaning that the S-ring arises as the transitivity module of a permutation group cont…

Let $G$ be a finite abelian group, and regard its underlying set with the monoid structure obtained by adjoining an identity in the semigroup construction denoted $G^e$. The number…

Klin's conjecture. Every S-ring over a cyclic group is Schurian.

Eventual stability conjecture. As $p \to \infty$, the set of Schur partitions of $G$ is eventually constant, and for sufficiently large $p$, every rational S-ring over $G$ has a ba…