Free group of Hamel bijections of big size

TL;DR

This paper constructs a large free group of Hamel functions of size continuum, including the identity, assuming the Continuum Hypothesis, addressing a recent open question in the field.

Contribution

It provides the first known construction of a free group of Hamel functions of maximal size under CH, expanding understanding of the algebraic structure of these functions.

Findings

Constructed a free group of size 2^c of Hamel functions

Included the identity function in the free group

Answered a recent open question about the existence of such groups

Abstract

A $f\colon\mathbb{R}\to\mathbb{R}$ is called Hamel function if its graph is a Hamel basis of the linear space $\mathbb{R}^2$ over rationals. We construct, assuming CH, a free group of the size $2^\mathfrak{c}$ contained in the class of all Hamel functions, with the indentity function included. This answers, consistently, a question posed recently by M. Lichman, M. Pawlikowski, S. Smolarek, and J. Swaczyna.