Topology of unitary groups and the prime orders of binomial coefficients

Let $c:SU(n)\rightarrow PSU(n)=SU(n)/\mathbb{Z}_{n}$ 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