On The Group Of Self-homotopy Equivalences Of An Elliptic Space

Let $X$ be a simply connected rational elliptic space of formal dimension $n$ and let $\text{E}(X)$ denote the group of homotopy classes of self-equivalences of $X$. If $X^{[k]}$ denotes the $k^{\text{th}}$ Postikov section of $X$ and $X^k$ denotes its $k^{\text{th}}$ skeleton, then making use of the models of Sullivan and Quillen we prove that $\text{E}(X)\cong \text{E}(X^{[n]})$ and if $n>m=max\{k\mathrm{\mid }{\pi }_k(X)\mathrm{\ne }0\}$ and $\text{E}(X)$ is finite, then $\text{E}(X)\cong \text{E}(X^{m+1})$. Moreover, in case when $X$ is 2-connected, we show that if ${\pi }_n(X)\mathrm{\ne }0$, then the group $\text{E}(X)$ is infinite.