Base sizes of primitive groups: bounds with explicit constants

Halasi, Zoltán and Liebeck, M.W. and Maróti, Attila (2019) Base sizes of primitive groups: bounds with explicit constants. JOURNAL OF ALGEBRA, 521. pp. 16-43. ISSN 0021-8693

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Abstract

We show that the minimal base size $b(G)$ of a finite primitive permutation group $G$ of degree $n$ is at most $2 (\log |G|/\log n) + 24$. This bound is asymptotically best possible since there exists a sequence of primitive permutation groups $G$ of degrees $n$ such that $b(G) = \lfloor 2 (\log |G|/\log n) \rceil - 2$ and $b(G)$ is unbounded. As a corollary we show that a primitive permutation group of degree $n$ that does not contain the alternating group $\mathrm{Alt}(n)$ has a base of size at most $\max\{\sqrt{n} , \ 25\}$.

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