Tending to a lecture on homotopy theory, the following question occured to me (is that a correct sentence?): 


Given a pointed space $(X,x)$, is the UNREDUCED suspension map $S:\pi\_k(X,x) \rightarrow \pi\_{k+1}(SX, \ast)$ a group homomorphism? 


Here unreduced suspension refers to $SX = X \times D^1 / \sim$, where $\sim$ collapses $X \times \{1\}$ and $X \times \{-1\}$ respectively, and the basepoint $\ast$ is (the one point set with element) $(x,0)$. 


The statement is contained in every book on homotopy theory and almost trivial for the REDUCED suspension. For wellpointed spaces this of course surfices to answer my question, but it has resisted several similar 'general nonsense'-arguments in the general case.



Unreduced Suspension Isomorphism
You can reduce to the well pointed case by observing that every space $X$ admits a weak equivalence $X'\to X$ from a well pointed space. Now the composition
$$
\pi\_kX'\to \pi\_kX\to \pi\_{k+1}SX
$$
(composition of your map with an isomorphism) is the same as the composition
$$
\pi\_kX'\to \pi\_{k+1}SX'\to \pi\_{k+1}SX
$$
in which the first map is a homomorphism because $X'$ is well pointed and the second is a homomorphism because it is induced by a map of spaces


