On a Conjecture of a Bound for the Exponent of the Schur Multiplier of a Finite \(p\)-Group

Let \(G\) be a \(p\)-group of nilpotency class \(k\) with finite exponent \(\exp(G)\) and let \(m=\lfloor\log_pk\rfloor\). We show that \(\exp(M^{(c)}(G))\) divides \(\exp(G)p^{m(k-1)}\), for all \(c\geq1\), where \(M^{(c)}(G)\) denotes the c-nilpotent multiplier of \(G\). This implies that \(\exp(M...

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Published in:arXiv.org 2010-11
Main Authors: Mashayekhy, Berooz, Hokmabadi, Azam, Mohammadzadeh, Fahimeh
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Mohammadzadeh, Fahimeh
description Let \(G\) be a \(p\)-group of nilpotency class \(k\) with finite exponent \(\exp(G)\) and let \(m=\lfloor\log_pk\rfloor\). We show that \(\exp(M^{(c)}(G))\) divides \(\exp(G)p^{m(k-1)}\), for all \(c\geq1\), where \(M^{(c)}(G)\) denotes the c-nilpotent multiplier of \(G\). This implies that \(\exp(M(G))\) divides \(\exp(G)\) for all finite \(p\)-groups of class at most \(p-1\). Moreover, we show that our result is an improvement of some previous bounds for the exponent of \(M^{(c)}(G)\) given by M. R. Jones, G. Ellis and P. Moravec in some cases.
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title On a Conjecture of a Bound for the Exponent of the Schur Multiplier of a Finite \(p\)-Group
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