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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Schur multipliers of some finite nilpotent groups


Author: David A. Jackson
Journal: Proc. Amer. Math. Soc. 66 (1977), 1-5
MSC: Primary 20D15
DOI: https://doi.org/10.1090/S0002-9939-1977-0460459-8
MathSciNet review: 0460459
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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\vert 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)$ .


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DOI: https://doi.org/10.1090/S0002-9939-1977-0460459-8
Keywords: Schur multiplier, Burnside group, commutator calculus
Article copyright: © Copyright 1977 American Mathematical Society