Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on vector space V, is not metacyclic. Then always has regular orbit V except for few “small” cases. We completely classify these cases in this paper.

Let $G$ be a permutation group on set $\Omega$. A base for is subset of $\Omega$ whose pointwise stabiliser trivial, and the size minimal cardinality base. If has $2$, then corresponding Saxl graph $\Sigma(G)$ vertex two vertices are adjacent if they form $G$. recent conjecture Burness Giudici states that finite primitive with property every have common neighbour. We investigate this when an af...

In this paper we show that if $G$ is a primitive subgroup of $S_{n}$ not large base, then any irredundant base for has size at most $5 \log n$. This the first logarithmic bound on an such groups, and best possible up to small constant. As corollary, relational complexity n+1$, maximal minimal height are both n.$ Furthermore, deduce n$ can be computed in polynomial time.