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Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the Pontryagin product is. The homology $H_*(C_{p^N};k)$ is a Hopf algebra that is dual to the cohomology Hopf algebra $H^*(C_{p^N};k)$, so this could be determined from the coalgebra structure on group cohomology.

In the case $p^N=2$, we have $BC_2\simeq \mathbb{RP}^\infty$, and it is well-known the Pontryagin product makes $H_*(C_2;k)$ a divided power algebra. This is because it is dual to the cohomology which is a polynomial ring. This is documented, for instance, in Hatcher in Example 3C.11 and the paragraph thereafter.

If $p^N>2$, then the cohomology has a more complicated algebra structure. It is well-known that $$ H^*(C_{p^N};k) \cong \Lambda[e] \otimes k[x], $$ where $\Lambda[e]$ denotes an exterior algebra, and the degrees are given by $|e|=1, |x|=2$. I have not been able to find any description of the coalgebra structure, however. Is there a reference which treats the coalgebra structure on cohomology or the Pontryagin product on homology in this case?

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    $\begingroup$ The coalgebra structure is determined by the coproduct on $e$ and $x$. $e$ is clearly primitive by degree reasons, and, when $p$ is odd, so is $x$: its coproduct, viewed as an element of $H^2(C \times C)$ must be invariant under swapping the two copies of $C$, and this eliminates the possibility of $e \otimes e$ being part of the formula. (When $p=2$ more care is needed.) $\endgroup$ Nov 27, 2023 at 22:05

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I recommend always looking at the canonical reference: Ken Brown's "Cohomology of Groups". Here Chapter V.5 is literally titled "The Pontraygin product" and then the very next Chapter V.6 is the application to abelian groups: Theorem V.6.6 $H_* \cong \Lambda_*\otimes\Gamma_*$ (exterior-algebra tensor divided-powers-algebra) holds even beyond cyclic.

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    $\begingroup$ Thanks! I feel foolish now for not looking there. $\endgroup$
    – Chase
    Nov 27, 2023 at 20:33

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