Groups of central type and Schur multipliers with large exponent. (Q1409610)

Let \(G\) be a finite group, \(M(G)=H_2(G,\mathbb{Z})\) the Schur multiplier of \(G\) and \(e\) the exponent of \(M(G)\). By an old result of Schur, \(e^2\) is a divisor of \(| G|\). The main result of this paper reads as follows. (a) \(e^2\) is a divisor of \(| G:Z^*(G)|\), where \(Z^*(G)\) is the characteristic subgroup of \(G\) which is minimal subject to being the image in \(G\) of the center of some central extension of \(G\); (b) If \(e^2=| G:Z^*(G)|\), then \(M(G)\) is cyclic of order \(e\) and \(G'\) is metabelian with \(Z^*(G')=1\) and with \(M(G')\) being isomorphic to the \(\pi(G')\) component of \(M(G)\). The proof relies among other things on the fact that in (b) a Schur cover \(E\) of \(G\) is of central type, that is, it has an irreducible character \(\chi\) such that \(\chi(1)^2=| E:Z(E)|\).