**Theorem.** *The Hopf invariant is a non-zero group homomorphism $\mathcal H:\pi_{4n-1}(\mathbb S^{2n})\to\mathbb Z$ and it is an isomorphism only when $n=1$.*

For a proof that $\mathcal H$ is a group homomorphism,
see Proposition 4B.1 in [3].
Hopf, in [4] (see Satz II and Satz II') proved that for any $n$, there is a map
$h:\mathbb S^{4n-1}\to\mathbb S^{2n}$ with $\mathcal H h\neq 0$ and hence the homomorphism
$\mathcal H:\pi_{4n-1}(\mathbb S^{2n})\to\mathbb Z$ is non-zero.
Since the Hopf invariant of the Hopf fibration $h:\mathbb S^3\to\mathbb S^2$ equals $1$ (Example 17.23 in [2]),
$\mathcal H:\pi_3(\mathbb S^2)\to\mathbb Z$ is an isomorphism. However, for $n\geq 2$ the Hopf invariant is never an isomorphism.
Indeed, Adams [1] proved that mappings with Hopf invariant equal $1$ exist only when $n=1,2$ and $4$,
so these are the only cases when one may suspect $\mathcal H$ to be an isomorphism, but
$\pi_7(\mathbb S^4)=\mathbb Z\times\mathbb Z_{12}$ and $\pi_{15}(\mathbb S^8)=\mathbb Z\times\mathbb Z_{120}$, so $\mathcal H$ cannot be an isomorphism.

[1] J. F. Adams, J.,
On the non-existence of elements of Hopf invariant one.
*Ann. of Math.* 72 (1960), 20-104.

[2] R. Bott, L. W. Tu, L. W.
*Differential forms in algebraic topology.*
Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982.

[3] A. Hatcher,
*Algebraic topology.*
Cambridge University Press, Cambridge, 2002.

[4] H. Hopf,
Über die Abbildungen von Sphären auf Sphären niedrigerer Dimensionen.
*Fundamenta Math.* 25 (1935), 427-440.

The topology of fibre bundles. $\endgroup$