First I'll construct the heat kernel measure $\mu _t$ on an infinite dimensional complex group $G$ using a diffusion in a Hilbert space. Using properties of this diffusion I can prove that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, $\mathcal{H}L^2(G, \mu _t)$, is one of two spaces of holomorphic functions I consider. Also I construct a subgroup $G_{CM}$ of $G$ which is an analog of the Cameron-Martin subspace. I'll describe an isometry from the first space to another space of holomorphic functions $\mathcal{H}L^2(G_{CM})$. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from $\mathcal{H}L^2(G_{CM})$ into the Hilbert space associated with a certain Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. I'll explain how my results on the Cameron-Martin subgroup are related to the work of I. Shigekawa and H. Sugita on the complex Wiener space.