#### Volume 17, issue 2 (2017)

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Hopf ring structure on the mod $p$ cohomology of symmetric groups

### Lorenzo Guerra

Algebraic & Geometric Topology 17 (2017) 957–982
##### Abstract

We describe a Hopf ring structure on ${\oplus }_{n\ge 0}{H}^{\ast }\left({\Sigma }_{n};{ℤ}_{p}\right)$, discovered by Strickland and Turner, where ${\Sigma }_{n}$ is the symmetric group of $n$ objects and $p$ is an odd prime. We also describe an additive basis on which the cup product is explicitly determined, compute the restriction to modular invariants and determine the action of the Steenrod algebra on our Hopf ring generators. For $p=2$ this was achieved in work of Giusti, Salvatore and Sinha, of which this work is an extension.

##### Keywords
group cohomology, symmetric group, Hopf ring, Dyer–Lashof operations, Steenrod algebra, Mui invariants
Primary: 20J06