On the size of the Schur multiplier of finite groups

We obtain bounds for the size of the Schur multiplier of finite $p$-groups and finite groups, which improve all existing bounds. Moreover, we obtain bounds for the size of the second cohomology group $H^2(G,\mathbb{Z}\mathrm{/}p\mathbb{Z})$ of a $p$-group with coefficients in $\mathbb{Z}\mathrm{/}p\mathbb{Z}$. Denoting the minimal number of generators of a $p$-group $G$ by $d(G)$, our bound depends on the parameters $\mathrm{\mid }G\mathrm{\mid }=p^n$, $\mathrm{\mid }{\gamma }_{2}G\mathrm{\mid }=p^k$, $d(G)=d$, $d(G\mathrm{/}Z)=\delta$ and $d({\gamma }_{2}G\mathrm{/}{\gamma }_{3}G)=k^{\mathrm{\prime }}$. For special $p$-groups, we further improve our bound when $\delta -1>k^{\mathrm{\prime }}$. Moreover, given natural numbers $d$, $\delta$, $k$ and $k^{\mathrm{\prime }}$ satisfying $k=k^{\mathrm{\prime }}$ and $\delta -1\le k^{\mathrm{\prime }}$, we construct a capable $p$-group $H$ of nilpotency class two and exponent $p$ such that the size of the Schur multiplier attains our bound.