Abstract

The covering number of a group G, denoted by $\sigma (G),$ is the size of a minimal collection of proper subgroups of G whose union is G. We investigate which integers are covering numbers of groups. We determine which integers 129 or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most 129 by introducing effective new computational techniques. Furthermore, we prove that, if ${\mathrm{F}}_1$ is the family of finite groups G such that all proper quotients of G are solvable, then $\mathbb{N}-\{\sigma (G):G\in {\mathrm{F}}_1\}$ is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form $(q^m-1)/(q-1),$ where $m\ne 3$ and q is a prime power, is a covering number, generalizing a result of Cohn.

摘要

组G的覆盖数，记为$\sigma (G),$是G的并集为G的适当子群的最小集合的大小。我们调查哪些整数覆盖了组数。我们确定了哪些整数 129 或更小是覆盖数，并且我们通过引入有效的新计算技术，精确地确定或限​​制了每个原始单片组的原始度最多为 129 的覆盖数。此外，我们证明，如果${\mathrm{F}}_1$是有限群G的族，使得G的所有真商都是可解的，则$\mathbb{ñ}-\{\sigma (G)：G\in {\mathrm{F}}_1\}$是无限的，这进一步证明了无限多的整数不能覆盖数字。最后，我们证明形式的每个整数$(q^{米}-1)/(q-1),$在哪里$米\ne 3$q是素数幂，是覆盖数，概括了 Cohn 的结果。