Suppose $k$ is a finite field, and $n,m\geq 4$ multiples of the field characteristic. Then the $A_\infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $H^*(C_m,k)$ are well known. We give results characterizing an $A_\infty$-structure on $H^*(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.
Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 1-11