A quotient group of the group of self homotopy equivalences of SO(4) (Q2427461)

For a pointed space \(X\), let \(\mathcal{E}(X)\) be the group of pointed homotopy classes of pointed self homotopy equivalences of \(X\) and let \(\mathcal{E}_{\#}(X)\) be the subgroup of \(\mathcal{E}(X)\) consisting of elements that induce the identity on homotopy groups. In the papers [\textit{A. J. Sieradski}, Pac. J. Math. 34, 789--802 (1970; Zbl 0194.55301) and \textit{K. Yamaguchi}, Hiroshima Math. J. 30, No.~1, 129--136 (2000; Zbl 0957.55005)], the quotient group \(\mathcal{E}(X)/\mathcal{E}_{\#}(X)\) has been determined for \(X= \text{SO}(4)\). In this article, the author proves that the order of any element in the group \(\mathcal{E}(\text{SO}(4))/\mathcal{E}_{\#}(\text{SO}(4))\) is \(1,2,4\) or \(\infty\).