Schur multipliers of some finite nilpotent groups

Abstract

Let B denote the Burnside group, B ( p α , d ) B({p^\alpha },d) and let G = B / B k G = B/{B_k} where p is a prime and 1 > k > p 1 > k > p . We show that the Schur multiplier, M ( G ) M(G) , is a direct power of Ψ ( k , d ) \Psi (k,d) cyclic groups, each having order p α {p^\alpha } , where Ψ ( k , d ) = k − 1 Σ n | k μ ( k / n ) d n \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 {F_k}/{F_{k + 1}} where F is free on d generators.) In addition we can show that M ( B ( 3 , d ) ) M(B(3,d)) is elementary abelian of exponent 3 and rank 2 ( 2 d ) + 4 ( 3 d ) + 3 ( 4 d ) 2(_2^d) + 4(_3^d) + 3(_4^d) .