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 $H^2(G,Z/pZ)$ of a p-group with coefficients in $Z/pZ$. Denoting the minimal number of generators of a p-group G by $d\left(G\right)$, our bound depends on the parameters $\left|G\right|=p^n$, $\left|{\gamma }_{2}G\right|=p^k$, $d\left(G\right)=d$, $d(G/Z)=\delta$ and $d({\gamma }_{2}G/{\gamma }_{3}G)=k^{\prime }$. For special p-groups, we further improve our bound when $\delta -1>k^{\prime }$. Moreover, given natural numbers d, δ, k and $k^{\prime }$ satisfying $k=k^{\prime }$ and $\delta -1\le k^{\prime }$, 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.