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In this paper, we establish a conjugacy theorem of homomorphisms of a locally compact connected semisimple group into a locally compact group.

Let $\mathscr{S}_{r}$, be the real topological vector space of real-valued rapidly decreasing functions and let $\mathcal{O}(\mathscr{S}_{r})$ be the group of rotations of $\mathscr{S}_{r}$. Then every one-parameter subgroup of $\mathcal{O}(\mathscr{S}_{r})$ induces a flow in $\mathscr{S}_{r}^{*}$ the conjugate space of $\mathscr{S}_{r}$ with the Gaussian White Noise as an invariant measure. ¶ The author constructed a group of functions which is isomorphic to a subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and some of its one-parameter subgroups. ¶ But the problem whether it contains sufficiently many one-parameter subgroups has been a problem. In Part I of the present paper, we answer this problem affirmatively by constructing two classes of one-parameter subgroups in a concrete way. ¶ In Part II, we construct an infinite dimensional Lie subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and the corresponding Lie algebra. Namely, we construct a topological subgroup $\mathfrak{G}$ of $\mathcal{O}(\mathscr{S}_{r})$ which is coordinated by the nuclear space $\mathscr{S}_{r}$ and the algebra $\mathfrak{a}$ of generators of one-parameter subgroups of $\mathfrak{G}$ which is closed under the commutation. Furthermore, we establish the exponential map from $\mathfrak{a}$ into $\mathfrak{G}$ and prove continuity.