On Modular maximal-cyclic braces

TL;DR

This paper investigates braces with modular maximal-cyclic adjoint groups, revealing they have a unique non-cyclic additive structure and that their count grows with group order, contrasting previous cases.

Contribution

It classifies braces with modular maximal-cyclic adjoint groups and shows their number increases with order, unlike other studied cases.

Findings

Braces with modular maximal-cyclic groups have a unique non-cyclic additive structure.

The number of such braces increases as the group order grows.

Contrasts previous cases where the number stabilized.

Abstract

Inspired by a conjecture by Guarnieri and Vendramin concerning the number of braces with a generalized quaternion adjoint group, many researchers have studied braces whose adjoint group is a non-abelian $2$-group with a cyclic subgroup of index $2$. Following this direction, braces with generalized quaternion, dihedral, and semidihedral adjoint groups have been classified. It was found that the number of such braces stabilizes as the group order increases. In this paper, we consider the remaining open case of modular maximal-cyclic groups. We show that these braces possess only one non-cyclic additive group structure, and, in contrast to previous findings, the number of such braces increases with increasing order.