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In this paper we give a complete isomorphical classification of free topological groups $FM\left(X\right)$ of locally compact zero-dimensional separable metric spaces $X$. From this classification we obtain for locally compact zero-dimensional separable metric spaces $X$ and $Y$ that the free topological groups $FM\left(X\right)$ and $FM\left(Y\right)$ are isomorphic if and only if $C_p\left(X\right)$ and $C_p\left(Y\right)$ are linearly homeomorphic.

Arhangel’skiǐ proved that if $X$ and $Y$ are completely regular spaces such that $C_p\left(X\right)$ and $C_p\left(Y\right)$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $\sigma$-compactness and realcompactness. In this paper we prove that if $\phi :C_p\left(X\right)\to C_p\left(X\right)$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $\phi$ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements do not hold...