Definably simple groups in o-minimal structures

Let $\mathbb{G} =\langle G, \cdot\rangle$ be a group definable in an o-minimal structure $\mathcal{M}$. A subset $H$ of $G$ is $\mathbb{G}$-definable if $H$ is definable in the structure $\langle G,\cdot\rangle$(while definable means definable in the structure $\mathcal{M}$). Assume $\mathbb{G}$ has no $\mathbb{G}$-definable proper subgroup of finite index. In this paper we prove that if $\mathbb{G}$has no nontrivial abelian normal subgroup, then $\mathbb{G}$ is the direct product of $\mathbb{G}$-definable subgroups $H_1,\ldots,H_k$ such that each $H_i$ is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.