20.126 (2022)

A brace $(G; +, \circ)$ is non-empty set $G$ with two binary operations $+$, $\circ$ such that $(G, +)$ is an additively written abelian group, $(G, \circ)$ is a multiplicatively written group, and $a \circ (b + c) + a = (a \circ b) + (a \circ c) for all a, b, c \in G$. Does there exist a brace with finitely generated group $(G, +)$ such that
$\qquad$ a) the group $(G, \circ)$ is non-solvable?
$\qquad$ b) the group $(G, \circ)$ contains a non-abelian free group?