On Left Braces in Which Every Subbrace is an Ideal

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 an elementary abelian p-group are also shown. As a consequence, every multipermutational Dedekind left brace whose additive group is an elementary abelian p-group is multipermutational of level 2. A new class of left braces, the extraspecial left braces, is introduced and play a prominent role in our approach.