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The Bockstein homomorphism can be generalized for $\mathbb{Z}_n$ values, $$\beta_n: H^m(M^d,\mathbb Z_n) \to H^{m+1}(M^d,\mathbb Z_n),$$ and $$\beta_n x =\frac1n d x \text{ mod } n,$$ $$x \in H^m(M^d,\mathbb Z_n),$$ $$\beta_n x \in H^{m+1}(M^d,\mathbb Z_n).$$

When $n=2$ for $\mathbb{Z}_2$, we can study the relation between Bockstein homomorphism and Steenrod square.

For general $\mathbb{Z}_n$, can we find any meaning between this Bockstein homomorphism and "Steenrod" $n$th power?

Is there such an "Steenrod" object to represent the $\beta_n x$?

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    If you read about the Steenrod algebra, you'll find that when $p$ is an odd prime, there are some relations between the Steenrod powers $P^n$ and the Bokstein homomorphism. These are called the Adem relations. – 2017-01-26
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    Thanks. This is the related case for Bockstein homomorphism and Steenrod square: http://math.stackexchange.com/questions/2113909/ What is the $n$ in your set-up? – 2017-01-27
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    @Jack Davies, do you mean Adem relations or Adams relations? – 2018-08-20

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