Groups of self-equivalences of suspended real projective spaces

The group consisting of the based homotopy classes of self-homotopy equivalences is called the self-equivalence group. We determine the group structures of self-equivalence groups, for the suspended real projective space whose dimension is less than or equal to six. The method is to study the multiplicative structure of self-homotopy set induced from the composition of maps. Finding out the invertible element of this rnonoid give almost all structures of self-equivalence groups. The group of the 1-fold suspension of the four-dimensional real projective space which is not determined similarly is obtained by the another method thought of from Rutter's paper.