Let $G$ be a group, and let $H$ be (the/a) maximal free subgroup of $G$.
Q1) Is the maximal free subgroup unique?
Q2) If $G$ has $m$ non-trivial generators then $H$ can have at most $m$ non-trivial generators. Is that true?
Thanks. I am new to free groups.
So far the examples that I can think of seems that it is true. For e.g. $G=\mathbb{Z}*(\mathbb{Z}/2)*(\mathbb{Z}/3)$, generated by 3 generators, then $H$=$\mathbb{Z}$, with 1 generator.