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A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is shown. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set by Tsang at Ischia Group Theory 2024 Conference is provided., Comment: 9 pages

The aim of this paper is to introduce and study the class of all left braces in which every subbrace is an ideal. We call them Dedekind left braces. It is proved that every finite Dedekind left brace is centrally nilpotent. Structural results about Dedekind left braces and a complete description of those ones whose additive group is elementary abelian are also shown. As a consequence, every hypermultipermutational Dedekind left brace whose additive group is elementary abelian is multipermutational of level $2$. A new class of left braces, the extraspecial left braces, is introduced and plays a prominent role in our approach., Comment: 25 pages

We show that the number of isomorphism classes of left braces of order~$64$ with additive group isomorphic to $C_2\times C_2\times C_4\times C_4$ is $10\,326\,821$. This completes the classification of left braces of order~$64$, that turn out to fall into $15\,095\,601$ isomorphism classes., Comment: Ric. Mat., in press. 9 pages. arXiv admin note: substantial text overlap with arXiv:2210.12823

We show that the number of isomorphism classes of left braces of order 64 with additive group isomorphic to $C_4 \times C_4 \times C_4$ is 1\,515\,429., Comment: Math. Comp., in press

We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$., Comment: 20 pages There is a change in the title with respect to the first draft. This paper has been published under an open access license thanks to the CRUE-CSIC agreement with Springer Nature

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability., Comment: 18 pages