Topology of unitary groups and the prime orders of binomial coefficients

Let $c:SU(n)\to PSU(n)=SU(n)\mathrm{/}{\mathbb{Z}}_n$ be the quotient map of the special unitary group $SU(n)$ by its center subgroup ${\mathbb{Z}}_n$. We determine the induced homomorphism $c^{\ast }:$ $H^{\ast}(PSU(n))\rightarrow H^{\ast}(SU(n))$ on cohomologies by computing with the prime orders of binomial coefficients