  Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free
topological group and the free Abelian topological group over $X$ in the sense
of Markov. In this paper, we consider two topological properties of $F(X)$ or
$A(X)$, namely the countable tightness and $\mathfrak G$-base. We provide some
characterizations of the countable tightness and $\mathfrak G$-base of $F(X)$
and $A(X)$ for various special classes of spaces $X$. Furthermore, we also
study the countable tightness and $\mathfrak G$-base of some $F_{n}(X)$ of
$F(X)$.
