Schur multipliers of some finite nilpotent groups

Abstract:

Let B denote the Burnside group, $B({p^\alpha },d)$ and let $G = B/{B_k}$ where p is a prime and $1 < k < p$. We show that the Schur multiplier, $M(G)$, is a direct power of $\Psi (k,d)$ cyclic groups, each having order ${p^\alpha }$, where $\Psi (k,d) = {k^{ - 1}}{\Sigma _{n|k}}\mu (k/n){d^n}$. (This is Witt’s formula for the rank of ${F_k}/{F_{k + 1}}$ where F is free on d generators.) In addition we can show that $M(B(3,d))$ is elementary abelian of exponent 3 and rank $2(_2^d) + 4(_3^d) + 3(_4^d)$ .