Steenrod reduced power ($P^i:H^q(X;{\mathbb{F}}_p)\to H^{q+2i(p-1)}(X;{\mathbb{F}}_p)$)

For odd prime $p$, the reduced powers are natural stable cohomology operations. They satisfy $P^0=1$, the Cartan formula, and the instability conditions $P^{i}a=0$ for $2i>|a|$ and $P^{i}a=a^p$ for $|a|=2i$. In particular, a degree-two class has $P^{1}x=x^p$ and no higher nonzero reduced power. Stability lets these operations detect distinctions between suspensions whose additive cohomology and cup products agree.

Choose the degree-four generator with pullback $u\mapsto x^2$ under the complex inclusion into quaternionic projective space. On infinite complex projective space, the Cartan formula gives $P^i(x^{2k})=\left(\genfrac{}{}{0ex}{}{2k}{i}\right)x^{2k+i(p-1)}$. Injectivity of the infinite-space pullback proves the displayed formula for quaternionic projective space. Restriction to ${\mathbb{HP}}^n$ sets powers above $n$ to zero. Coefficients are reduced modulo the odd prime $p$; the exponent is an integer because $p-1$ is even.