= Stable cohomology operation
A <cohomology operation> is stable when it commutes with the <suspension isomorphism> on <reduced cohomology>:
$$
\theta(\sigma a)=\sigma(\theta a).
$$
Products of positive-degree reduced classes on a <suspension> vanish, but stable operations can still connect its nonzero <cohomology classes>. The <Steenrod squares> and, at odd primes, <Steenrod reduced powers> are examples.
Let $x\in H^2(\mathbb{CP}^2;\mathbb F_2)$ be the <hyperplane class> modulo two. It satisfies $\operatorname{Sq}^2x=x^2\ne0$. Stability gives $\operatorname{Sq}^2(\sigma x)=\sigma(x^2)\ne0$ on $\Sigma\mathbb{CP}^2$. The <suspension> of $S^2\vee S^4$ has the same additive <groups> and vanishing positive-degree products, but its corresponding degree-three class has zero $\operatorname{Sq}^2$, by <pullback> from $S^3$. Thus these suspensions have different <cohomology operations> even though their additive <groups> and <cup products> agree.<Algebraic Topology by Allen Hatcher>[section4.L, Steenrod squares and powers]{ref}
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