# What is the difference between multiplication and direct sum on homotopy groups of spheres?

In Allen Hatcher's book Algebraic topology he states that factoring out 2-torsion $\pi_{i}(S^{2n})\cong\pi_{i-1}(S^{2n-1})\times\pi_{i}(S^{4n-1})\:\forall n$ but in his book Spectral Sequences in Algebraic Topology he has $\pi_{i}(S^{n})\cong\pi_{i-1}(S^{n-1})\bigoplus\pi_{i}(S^{2n-1})\:\forall n$ even. Am I correct in assuming these are the same or am I totally missing something?

$$\prod^{n}_{i=1}A_{i}=\bigoplus^{n}_{i=1}A_{i}$$ by mapping $(a_{1}\cdots a_{n})$ to $\sum_{i=1}^{n}a_{i}$.
• No, I mean "commutative coproduct." The direct sum of a family of groups $G_i$ is the universal group equipped with a map from all of the $G_i$ all of whose images commute. – Qiaochu Yuan Mar 2 '14 at 19:41