Often in mathematics we define constructions outputting objects which *a priori* have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

- Given a Lie group , its tangent space at the identity is
*a priori*a vector space, but it ends up having the structure of a Lie algebra. - Given a space , its cohomology is
*a priori*a graded abelian group, but it ends up having the structure of a graded ring. - Given a space , its cohomology over is
*a priori*a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod- Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups of a space: that is, we will show that, in a suitable sense, each individual homotopy group is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)