Schur multipliers and power endomorphisms of groups.
A group \(G\) is said to be \(n\)-central if \(\exp(G/Z(G))\) divides \(n\). Clearly, if \(\exp(G)=n\), then a representation group of \(G\) is \(n\)-central. Using his results on \(n\)-central groups, the author proves some new estimates of \(\exp(M(G))\), where \(M(G)\) is the Schur multiplier of \(G\). As a consequence, he proves that if \(G\) is a metabelian group of exponent \(p\), then \(\exp(M(G))\) divides \(p\) (in fact, according to D. L. Johnson, we have \(\exp(M(G))=p\) here). Next, if \(\exp(G)=4\), then \(\exp(M(G))\) divides \(8\), and this is the best possible result.
scientific article; zbMATH DE number 5129703
scientific article