On the size of the Schur multiplier of finite groups

Abstract: 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 $H^2(G,\mathbb{Z}/p\mathbb{Z})$ of a $p$-group with coefficients in $\mathbb{Z}/p\mathbb{Z}$. Denoting the minimal number of generators of a $p$-group $G$ by $d(G)$, our bound depends on the parameters $|G|=p^n$, $|\gamma_2G|=p^k$, $d(G)=d$, $d(G/Z)=\delta$ and $d(\gamma_2G/\gamma_3G)=k'$. For special $p$-groups, we further improve our bound when $\delta-1 > k'$. Moreover, given natural numbers $d$, $\delta$, $k$ and $k'$ satisfying $k=k'$ and $\delta-1 \leq k'$, 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.