 Steenrod reduced power

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.