On the relation between upper central quotients and lower central series of a group (Q2723473)

There are many well-known theorems concerning the influence of the upper central quotient group \(H=G/Z_c(G)\) on the lower central subgroup \(\gamma_{c+1}(G)\). An old result of \textit{R. Baer} [Trans. Am. Math. Soc. 58, 295-347, 348-389, 390-419 (1945; Zbl 0061.02703)] states that \(\gamma_{c+1}(G)\) is finite whenever the quotient \(H=G/Z_c(G)\) is finite. In the article under review the author gives an upper bound on the order and exponent of \(\gamma_{c+1}(G)\). For \(H\) equal to a dihedral group, or a quaternion group, or an extra-special group all possible groups that can arise as \(\gamma_{c+1}(G)\) are listed. The proofs involve the Baer invariants, the Schur multiplier of a pairs of groups, and the nonabelian tensor product of groups. Bounds on the order and exponent of the Baer invariant \(M^{(c)}(H)\) for an arbitrary finite group \(H\) are also obtained. In particular, it is proved that the exponent of the Baer invariant \(M^{(c)}(P)\) of a finite \(p\)-group \(P\) of class \(k\) and exponent \(p^e\) divides \(p^{[k/2]e}\), where \([k/2]\) is the least integer greater or equal to \(k/2\) (and even divides \(p^e\) if \(P\) is a powerful \(p\)-group).