What is the $p$-primary component? I got stuck when reading Differential topology 46 years later in the last section of the article ("Further details"). It is a summary of what is known about stable homotopy groups of spheres $\Pi_n$. It is stated that "the most difficult part is the $2$-primary component" and also $p$-primary components are mentioned. 
What is a "$p$-primary component" and what does it have to do with a homotopy group of spheres? Recently I started reading different papers and articles about differential topology but it is the first time that I encounter this concept but it is not explained in the article.
 A: From Adams' Infinite Loop Spaces book: Let $X$ be a simply connected CW complex and let $A$ be a subring of the rational numbers. Then there is a simply connected $CW$-complex $Y$ and a universal map $i : X \rightarrow Y$ such that $i$ localizes homotopy, in the sense that $i_{*} : \pi_{r}(X) \rightarrow \pi_{r}(Y)$ induces an isomorphism $\pi_{r}(X) \otimes A \rightarrow \pi_{r}(Y)$. The term localization is used as this should remind you of localizations of modules. 
Now let $p$ be a prime and let $A_{p} \subset \mathbb{Q}$ be the subring of all rationals with denominator not a multiple of $p$. Then for any group $G$, the group $G \otimes A_{p}$ consists of the $p$-torsion elements of $G$ only. Thus if we localize a topological space at this ring we obtain a space which we call "$X$ localised at $p$" whose homotopy groups consist of the $p$-torsion parts of the homotopy groups of $X$. By working one prime at a time you can calculate the entire torsion part of the homotopy groups of $X$. Since we know that the stable homotopy groups of spheres are finite for all $r > 0$, this allows them to be calculated if we know the $p$-primary parts.
